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Display properties: number-header: tableau coefficient-Numbers: []+Numbers:+- params:+ method: forward-euler+ coefficient: b1_1+ number: '1'+ param-latex: $b^{(1)}_1$+ comment: 'Forward Euler method: $b^{(1)}=(1)$.'+- params:+ method: midpoint+ coefficient: c2+ number: 1/2+ param-latex: $c_2$+ comment: 'Explicit midpoint method: $b^{(2)}=(0, 1)$.'+- params:+ method: midpoint+ coefficient: a2_1+ number: 1/2+ param-latex: $a_{21}$+- params:+ method: midpoint+ coefficient: b2_2+ number: '1'+ param-latex: $b^{(2)}_2$+- params:+ method: heun-2+ coefficient: c2+ number: '1'+ param-latex: $c_2$+ comment: 'Heun''s second-order method: $b^{(2)}=(\tfrac{1}{2}, \tfrac{1}{2})$.'+- params:+ method: heun-2+ coefficient: a2_1+ number: '1'+ param-latex: $a_{21}$+- params:+ method: heun-2+ coefficient: b2_1+ number: 1/2+ param-latex: $b^{(2)}_1$+- params:+ method: heun-2+ coefficient: b2_2+ number: 1/2+ param-latex: $b^{(2)}_2$+- params:+ method: ralston-2+ coefficient: c2+ number: 2/3+ param-latex: $c_2$+ comment: 'Ralston''s second-order method: $b^{(2)}=(\tfrac{1}{4}, \tfrac{3}{4})$.'+- params:+ method: ralston-2+ coefficient: a2_1+ number: 2/3+ param-latex: $a_{21}$+- params:+ method: ralston-2+ coefficient: b2_1+ number: 1/4+ param-latex: $b^{(2)}_1$+- params:+ method: ralston-2+ coefficient: b2_2+ number: 3/4+ param-latex: $b^{(2)}_2$+- params:+ method: kutta-3+ coefficient: c2+ number: 1/2+ param-latex: $c_2$+ comment: 'Kutta''s third-order method: $b^{(3)}=(\tfrac{1}{6}, \tfrac{2}{3}, \tfrac{1}{6})$.'+- params:+ method: kutta-3+ coefficient: a2_1+ number: 1/2+ param-latex: $a_{21}$+- params:+ method: kutta-3+ coefficient: c3+ number: '1'+ param-latex: $c_3$+- params:+ method: kutta-3+ coefficient: a3_1+ number: '-1'+ param-latex: $a_{31}$+- params:+ method: kutta-3+ coefficient: a3_2+ number: '2'+ param-latex: $a_{32}$+- params:+ method: kutta-3+ coefficient: b3_1+ number: 1/6+ param-latex: $b^{(3)}_1$+- params:+ method: kutta-3+ coefficient: b3_2+ number: 2/3+ param-latex: $b^{(3)}_2$+- params:+ method: kutta-3+ coefficient: b3_3+ number: 1/6+ param-latex: $b^{(3)}_3$+- params:+ method: heun-3+ coefficient: c2+ number: 1/3+ param-latex: $c_2$+ comment: 'Heun''s third-order method: $b^{(3)}=(\tfrac{1}{4}, 0, \tfrac{3}{4})$.'+- params:+ method: heun-3+ coefficient: a2_1+ number: 1/3+ param-latex: $a_{21}$+- params:+ method: heun-3+ coefficient: c3+ number: 2/3+ param-latex: $c_3$+- params:+ method: heun-3+ coefficient: a3_2+ number: 2/3+ param-latex: $a_{32}$+- params:+ method: heun-3+ coefficient: b3_1+ number: 1/4+ param-latex: $b^{(3)}_1$+- params:+ method: heun-3+ coefficient: b3_3+ number: 3/4+ param-latex: $b^{(3)}_3$+- params:+ method: ralston-3+ coefficient: c2+ number: 1/2+ param-latex: $c_2$+ comment: 'Ralston''s third-order method: $b^{(3)}=(\tfrac{2}{9}, \tfrac{1}{3}, \tfrac{4}{9})$.'+- params:+ method: ralston-3+ coefficient: a2_1+ number: 1/2+ param-latex: $a_{21}$+- params:+ method: ralston-3+ coefficient: c3+ number: 3/4+ param-latex: $c_3$+- params:+ method: ralston-3+ coefficient: a3_2+ number: 3/4+ param-latex: $a_{32}$+- params:+ method: ralston-3+ coefficient: b3_1+ number: 2/9+ param-latex: $b^{(3)}_1$+- params:+ method: ralston-3+ coefficient: b3_2+ number: 1/3+ param-latex: $b^{(3)}_2$+- params:+ method: ralston-3+ coefficient: b3_3+ number: 4/9+ param-latex: $b^{(3)}_3$+- params:+ method: van-der-houwen-wray+ coefficient: c2+ number: 8/15+ param-latex: $c_2$+ comment: 'Van der Houwen-Wray third-order method: $b^{(3)}=(\tfrac{1}{4}, 0, \tfrac{3}{4})$.'+- params:+ method: van-der-houwen-wray+ coefficient: a2_1+ number: 8/15+ param-latex: $a_{21}$+- params:+ method: van-der-houwen-wray+ coefficient: c3+ number: 2/3+ param-latex: $c_3$+- params:+ method: van-der-houwen-wray+ coefficient: a3_1+ number: 1/4+ param-latex: $a_{31}$+- params:+ method: van-der-houwen-wray+ coefficient: a3_2+ number: 5/12+ param-latex: $a_{32}$+- params:+ method: van-der-houwen-wray+ coefficient: b3_1+ number: 1/4+ param-latex: $b^{(3)}_1$+- params:+ method: van-der-houwen-wray+ coefficient: b3_3+ number: 3/4+ param-latex: $b^{(3)}_3$+- params:+ method: ssprk3+ coefficient: c2+ number: '1'+ param-latex: $c_2$+ comment: 'Third-order strong stability preserving Runge-Kutta method: $b^{(3)}=(\tfrac{1}{6},+ \tfrac{1}{6}, \tfrac{2}{3})$.'+- params:+ method: ssprk3+ coefficient: a2_1+ number: '1'+ param-latex: $a_{21}$+- params:+ method: ssprk3+ coefficient: c3+ number: 1/2+ param-latex: $c_3$+- params:+ method: ssprk3+ coefficient: a3_1+ number: 1/4+ param-latex: $a_{31}$+- params:+ method: ssprk3+ coefficient: a3_2+ number: 1/4+ param-latex: $a_{32}$+- params:+ method: ssprk3+ coefficient: b3_1+ number: 1/6+ param-latex: $b^{(3)}_1$+- params:+ method: ssprk3+ coefficient: b3_2+ number: 1/6+ param-latex: $b^{(3)}_2$+- params:+ method: ssprk3+ coefficient: b3_3+ number: 2/3+ param-latex: $b^{(3)}_3$+- params:+ method: rk4+ coefficient: c2+ number: 1/2+ param-latex: $c_2$+ comment: 'Classic fourth-order Runge-Kutta method: $b^{(4)}=(\tfrac{1}{6}, \tfrac{1}{3},+ \tfrac{1}{3}, \tfrac{1}{6})$.'+- params:+ method: rk4+ coefficient: a2_1+ number: 1/2+ param-latex: $a_{21}$+- params:+ method: rk4+ coefficient: c3+ number: 1/2+ param-latex: $c_3$+- params:+ method: rk4+ coefficient: a3_2+ number: 1/2+ param-latex: $a_{32}$+- params:+ method: rk4+ coefficient: c4+ number: '1'+ param-latex: $c_4$+- params:+ method: rk4+ coefficient: a4_3+ number: '1'+ param-latex: $a_{43}$+- params:+ method: rk4+ coefficient: b4_1+ number: 1/6+ param-latex: $b^{(4)}_1$+- params:+ method: rk4+ coefficient: b4_2+ number: 1/3+ param-latex: $b^{(4)}_2$+- params:+ method: rk4+ coefficient: b4_3+ number: 1/3+ param-latex: $b^{(4)}_3$+- params:+ method: rk4+ coefficient: b4_4+ number: 1/6+ param-latex: $b^{(4)}_4$+- params:+ method: three-eighths+ coefficient: c2+ number: 1/3+ param-latex: $c_2$+ comment: '3/8-rule fourth-order method: $b^{(4)}=(\tfrac{1}{8}, \tfrac{3}{8}, \tfrac{3}{8},+ \tfrac{1}{8})$.'+- params:+ method: three-eighths+ coefficient: a2_1+ number: 1/3+ param-latex: $a_{21}$+- params:+ method: three-eighths+ coefficient: c3+ number: 2/3+ param-latex: $c_3$+- params:+ method: three-eighths+ coefficient: a3_1+ number: -1/3+ param-latex: $a_{31}$+- params:+ method: three-eighths+ coefficient: a3_2+ number: '1'+ param-latex: $a_{32}$+- params:+ method: three-eighths+ coefficient: c4+ number: '1'+ param-latex: $c_4$+- params:+ method: three-eighths+ coefficient: a4_1+ number: '1'+ param-latex: $a_{41}$+- params:+ method: three-eighths+ coefficient: a4_2+ number: '-1'+ param-latex: $a_{42}$+- params:+ method: three-eighths+ coefficient: a4_3+ number: '1'+ param-latex: $a_{43}$+- params:+ method: three-eighths+ coefficient: b4_1+ number: 1/8+ param-latex: $b^{(4)}_1$+- params:+ method: three-eighths+ coefficient: b4_2+ number: 3/8+ param-latex: $b^{(4)}_2$+- params:+ method: three-eighths+ coefficient: b4_3+ number: 3/8+ param-latex: $b^{(4)}_3$+- params:+ method: three-eighths+ coefficient: b4_4+ number: 1/8+ param-latex: $b^{(4)}_4$+- params:+ method: nystrom-5+ coefficient: c2+ number: 1/3+ param-latex: $c_2$+ comment: 'Nystrom''s fifth-order method: $b^{(5)}=(\tfrac{23}{192}, 0, \tfrac{125}{192},+ 0, -\tfrac{27}{64}, \tfrac{125}{192})$.'+- params:+ method: nystrom-5+ coefficient: a2_1+ number: 1/3+ param-latex: $a_{21}$+- params:+ method: nystrom-5+ coefficient: c3+ number: 2/5+ param-latex: $c_3$+- params:+ method: nystrom-5+ coefficient: a3_1+ number: 4/25+ param-latex: $a_{31}$+- params:+ method: nystrom-5+ coefficient: a3_2+ number: 6/25+ param-latex: $a_{32}$+- params:+ method: nystrom-5+ coefficient: c4+ number: '1'+ param-latex: $c_4$+- params:+ method: nystrom-5+ coefficient: a4_1+ number: 1/4+ param-latex: $a_{41}$+- params:+ method: nystrom-5+ coefficient: a4_2+ number: '-3'+ param-latex: $a_{42}$+- params:+ method: nystrom-5+ coefficient: a4_3+ number: 15/4+ param-latex: $a_{43}$+- params:+ method: nystrom-5+ coefficient: c5+ number: 2/3+ param-latex: $c_5$+- params:+ method: nystrom-5+ coefficient: a5_1+ number: 2/27+ param-latex: $a_{51}$+- params:+ method: nystrom-5+ coefficient: a5_2+ number: 10/9+ param-latex: $a_{52}$+- params:+ method: nystrom-5+ coefficient: a5_3+ number: -50/81+ param-latex: $a_{53}$+- params:+ method: nystrom-5+ coefficient: a5_4+ number: 8/81+ param-latex: $a_{54}$+- params:+ method: nystrom-5+ coefficient: c6+ number: 4/5+ param-latex: $c_6$+- params:+ method: nystrom-5+ coefficient: a6_1+ number: 2/25+ param-latex: $a_{61}$+- params:+ method: nystrom-5+ coefficient: a6_2+ number: 12/25+ param-latex: $a_{62}$+- params:+ method: nystrom-5+ coefficient: a6_3+ number: 2/15+ param-latex: $a_{63}$+- params:+ method: nystrom-5+ coefficient: a6_4+ number: 8/75+ param-latex: $a_{64}$+- params:+ method: nystrom-5+ coefficient: b5_1+ number: 23/192+ param-latex: $b^{(5)}_1$+- params:+ method: nystrom-5+ coefficient: b5_3+ number: 125/192+ param-latex: $b^{(5)}_3$+- params:+ method: nystrom-5+ coefficient: b5_5+ number: -27/64+ param-latex: $b^{(5)}_5$+- params:+ method: nystrom-5+ coefficient: b5_6+ number: 125/192+ param-latex: $b^{(5)}_6$+- params:+ method: heun-euler+ coefficient: c2+ number: '1'+ param-latex: $c_2$+ comment: 'Heun-Euler embedded pair: $b^{(2)}=(\tfrac{1}{2}, \tfrac{1}{2})$; $b^{(1)}=(1,+ 0)$. The order-2 row is the advancing formula.'+- params:+ method: heun-euler+ coefficient: a2_1+ number: '1'+ param-latex: $a_{21}$+- params:+ method: heun-euler+ coefficient: b2_1+ number: 1/2+ param-latex: $b^{(2)}_1$+- params:+ method: heun-euler+ coefficient: b2_2+ number: 1/2+ param-latex: $b^{(2)}_2$+- params:+ method: heun-euler+ coefficient: b1_1+ number: '1'+ param-latex: $b^{(1)}_1$+- params:+ method: fehlberg-12+ coefficient: c2+ number: 1/2+ param-latex: $c_2$+ comment: 'Fehlberg RK1(2) embedded pair: $b^{(2)}=(\tfrac{1}{512}, \tfrac{255}{256},+ \tfrac{1}{512})$; $b^{(1)}=(\tfrac{1}{256}, \tfrac{255}{256}, 0)$. The order-1+ row is the advancing formula.'+- params:+ method: fehlberg-12+ coefficient: a2_1+ number: 1/2+ param-latex: $a_{21}$+- params:+ method: fehlberg-12+ coefficient: c3+ number: '1'+ param-latex: $c_3$+- params:+ method: fehlberg-12+ coefficient: a3_1+ number: 1/256+ param-latex: $a_{31}$+- params:+ method: fehlberg-12+ coefficient: a3_2+ number: 255/256+ param-latex: $a_{32}$+- params:+ method: fehlberg-12+ coefficient: b2_1+ number: 1/512+ param-latex: $b^{(2)}_1$+- params:+ method: fehlberg-12+ coefficient: b2_2+ number: 255/256+ param-latex: $b^{(2)}_2$+- params:+ method: fehlberg-12+ coefficient: b2_3+ number: 1/512+ param-latex: $b^{(2)}_3$+- params:+ method: fehlberg-12+ coefficient: b1_1+ number: 1/256+ param-latex: $b^{(1)}_1$+- params:+ method: fehlberg-12+ coefficient: b1_2+ number: 255/256+ param-latex: $b^{(1)}_2$+- params:+ method: bogacki-shampine+ coefficient: c2+ number: 1/2+ param-latex: $c_2$+ comment: 'Bogacki-Shampine embedded pair: $b^{(3)}=(\tfrac{2}{9}, \tfrac{1}{3},+ \tfrac{4}{9}, 0)$; $b^{(2)}=(\tfrac{7}{24}, \tfrac{1}{4}, \tfrac{1}{3}, \tfrac{1}{8})$.+ The order-3 row is the advancing formula. This is SciPy''s RK23 tableau.'+- params:+ method: bogacki-shampine+ coefficient: a2_1+ number: 1/2+ param-latex: $a_{21}$+- params:+ method: bogacki-shampine+ coefficient: c3+ number: 3/4+ param-latex: $c_3$+- params:+ method: bogacki-shampine+ coefficient: a3_2+ number: 3/4+ param-latex: $a_{32}$+- params:+ method: bogacki-shampine+ coefficient: c4+ number: '1'+ param-latex: $c_4$+- params:+ method: bogacki-shampine+ coefficient: a4_1+ number: 2/9+ param-latex: $a_{41}$+- params:+ method: bogacki-shampine+ coefficient: a4_2+ number: 1/3+ param-latex: $a_{42}$+- params:+ method: bogacki-shampine+ coefficient: a4_3+ number: 4/9+ param-latex: $a_{43}$+- params:+ method: bogacki-shampine+ coefficient: b3_1+ number: 2/9+ param-latex: $b^{(3)}_1$+- params:+ method: bogacki-shampine+ coefficient: b3_2+ number: 1/3+ param-latex: $b^{(3)}_2$+- params:+ method: bogacki-shampine+ coefficient: b3_3+ number: 4/9+ param-latex: $b^{(3)}_3$+- params:+ method: bogacki-shampine+ coefficient: b2_1+ number: 7/24+ param-latex: $b^{(2)}_1$+- params:+ method: bogacki-shampine+ coefficient: b2_2+ number: 1/4+ param-latex: $b^{(2)}_2$+- params:+ method: bogacki-shampine+ coefficient: b2_3+ number: 1/3+ param-latex: $b^{(2)}_3$+- params:+ method: bogacki-shampine+ coefficient: b2_4+ number: 1/8+ param-latex: $b^{(2)}_4$+- params:+ method: fehlberg-45+ coefficient: c2+ number: 1/4+ param-latex: $c_2$+ comment: 'Runge-Kutta-Fehlberg RKF45 embedded pair: $b^{(5)}=(\tfrac{16}{135}, 0,+ \tfrac{6656}{12825}, \tfrac{28561}{56430}, -\tfrac{9}{50}, \tfrac{2}{55})$; $b^{(4)}=(\tfrac{25}{216},+ 0, \tfrac{1408}{2565}, \tfrac{2197}{4104}, -\tfrac{1}{5}, 0)$. The order-4 row+ is the advancing formula.'+- params:+ method: fehlberg-45+ coefficient: a2_1+ number: 1/4+ param-latex: $a_{21}$+- params:+ method: fehlberg-45+ coefficient: c3+ number: 3/8+ param-latex: $c_3$+- params:+ method: fehlberg-45+ coefficient: a3_1+ number: 3/32+ param-latex: $a_{31}$+- params:+ method: fehlberg-45+ coefficient: a3_2+ number: 9/32+ param-latex: $a_{32}$+- params:+ method: fehlberg-45+ coefficient: c4+ number: 12/13+ param-latex: $c_4$+- params:+ method: fehlberg-45+ coefficient: a4_1+ number: 1932/2197+ param-latex: $a_{41}$+- params:+ method: fehlberg-45+ coefficient: a4_2+ number: -7200/2197+ param-latex: $a_{42}$+- params:+ method: fehlberg-45+ coefficient: a4_3+ number: 7296/2197+ param-latex: $a_{43}$+- params:+ method: fehlberg-45+ coefficient: c5+ number: '1'+ param-latex: $c_5$+- params:+ method: fehlberg-45+ coefficient: a5_1+ number: 439/216+ param-latex: $a_{51}$+- params:+ method: fehlberg-45+ coefficient: a5_2+ number: '-8'+ param-latex: $a_{52}$+- params:+ method: fehlberg-45+ coefficient: a5_3+ number: 3680/513+ param-latex: $a_{53}$+- params:+ method: fehlberg-45+ coefficient: a5_4+ number: -845/4104+ param-latex: $a_{54}$+- params:+ method: fehlberg-45+ coefficient: c6+ number: 1/2+ param-latex: $c_6$+- params:+ method: fehlberg-45+ coefficient: a6_1+ number: -8/27+ param-latex: $a_{61}$+- params:+ method: fehlberg-45+ coefficient: a6_2+ number: '2'+ param-latex: $a_{62}$+- params:+ method: fehlberg-45+ coefficient: a6_3+ number: -3544/2565+ param-latex: $a_{63}$+- params:+ method: fehlberg-45+ coefficient: a6_4+ number: 1859/4104+ param-latex: $a_{64}$+- params:+ method: fehlberg-45+ coefficient: a6_5+ number: -11/40+ param-latex: $a_{65}$+- params:+ method: fehlberg-45+ coefficient: b5_1+ number: 16/135+ param-latex: $b^{(5)}_1$+- params:+ method: fehlberg-45+ coefficient: b5_3+ number: 6656/12825+ param-latex: $b^{(5)}_3$+- params:+ method: fehlberg-45+ coefficient: b5_4+ number: 28561/56430+ param-latex: $b^{(5)}_4$+- params:+ method: fehlberg-45+ coefficient: b5_5+ number: -9/50+ param-latex: $b^{(5)}_5$+- params:+ method: fehlberg-45+ coefficient: b5_6+ number: 2/55+ param-latex: $b^{(5)}_6$+- params:+ method: fehlberg-45+ coefficient: b4_1+ number: 25/216+ param-latex: $b^{(4)}_1$+- params:+ method: fehlberg-45+ coefficient: b4_3+ number: 1408/2565+ param-latex: $b^{(4)}_3$+- params:+ method: fehlberg-45+ coefficient: b4_4+ number: 2197/4104+ param-latex: $b^{(4)}_4$+- params:+ method: fehlberg-45+ coefficient: b4_5+ number: -1/5+ param-latex: $b^{(4)}_5$+- params:+ method: cash-karp+ coefficient: c2+ number: 1/5+ param-latex: $c_2$+ comment: 'Cash-Karp embedded pair: $b^{(5)}=(\tfrac{37}{378}, 0, \tfrac{250}{621},+ \tfrac{125}{594}, 0, \tfrac{512}{1771})$; $b^{(4)}=(\tfrac{2825}{27648}, 0, \tfrac{18575}{48384},+ \tfrac{13525}{55296}, \tfrac{277}{14336}, \tfrac{1}{4})$. The order-5 row is the+ advancing formula.'+- params:+ method: cash-karp+ coefficient: a2_1+ number: 1/5+ param-latex: $a_{21}$+- params:+ method: cash-karp+ coefficient: c3+ number: 3/10+ param-latex: $c_3$+- params:+ method: cash-karp+ coefficient: a3_1+ number: 3/40+ param-latex: $a_{31}$+- params:+ method: cash-karp+ coefficient: a3_2+ number: 9/40+ param-latex: $a_{32}$+- params:+ method: cash-karp+ coefficient: c4+ number: 3/5+ param-latex: $c_4$+- params:+ method: cash-karp+ coefficient: a4_1+ number: 3/10+ param-latex: $a_{41}$+- params:+ method: cash-karp+ coefficient: a4_2+ number: -9/10+ param-latex: $a_{42}$+- params:+ method: cash-karp+ coefficient: a4_3+ number: 6/5+ param-latex: $a_{43}$+- params:+ method: cash-karp+ coefficient: c5+ number: '1'+ param-latex: $c_5$+- params:+ method: cash-karp+ coefficient: a5_1+ number: -11/54+ param-latex: $a_{51}$+- params:+ method: cash-karp+ coefficient: a5_2+ number: 5/2+ param-latex: $a_{52}$+- params:+ method: cash-karp+ coefficient: a5_3+ number: -70/27+ param-latex: $a_{53}$+- params:+ method: cash-karp+ coefficient: a5_4+ number: 35/27+ param-latex: $a_{54}$+- params:+ method: cash-karp+ coefficient: c6+ number: 7/8+ param-latex: $c_6$+- params:+ method: cash-karp+ coefficient: a6_1+ number: 1631/55296+ param-latex: $a_{61}$+- params:+ method: cash-karp+ coefficient: a6_2+ number: 175/512+ param-latex: $a_{62}$+- params:+ method: cash-karp+ coefficient: a6_3+ number: 575/13824+ param-latex: $a_{63}$+- params:+ method: cash-karp+ coefficient: a6_4+ number: 44275/110592+ param-latex: $a_{64}$+- params:+ method: cash-karp+ coefficient: a6_5+ number: 253/4096+ param-latex: $a_{65}$+- params:+ method: cash-karp+ coefficient: b5_1+ number: 37/378+ param-latex: $b^{(5)}_1$+- params:+ method: cash-karp+ coefficient: b5_3+ number: 250/621+ param-latex: $b^{(5)}_3$+- params:+ method: cash-karp+ coefficient: b5_4+ number: 125/594+ param-latex: $b^{(5)}_4$+- params:+ method: cash-karp+ coefficient: b5_6+ number: 512/1771+ param-latex: $b^{(5)}_6$+- params:+ method: cash-karp+ coefficient: b4_1+ number: 2825/27648+ param-latex: $b^{(4)}_1$+- params:+ method: cash-karp+ coefficient: b4_3+ number: 18575/48384+ param-latex: $b^{(4)}_3$+- params:+ method: cash-karp+ coefficient: b4_4+ number: 13525/55296+ param-latex: $b^{(4)}_4$+- params:+ method: cash-karp+ coefficient: b4_5+ number: 277/14336+ param-latex: $b^{(4)}_5$+- params:+ method: cash-karp+ coefficient: b4_6+ number: 1/4+ param-latex: $b^{(4)}_6$+- params:+ method: dormand-prince+ coefficient: c2+ number: 1/5+ param-latex: $c_2$+ comment: 'Dormand-Prince embedded pair: $b^{(5)}=(\tfrac{35}{384}, 0, \tfrac{500}{1113},+ \tfrac{125}{192}, -\tfrac{2187}{6784}, \tfrac{11}{84}, 0)$; $b^{(4)}=(\tfrac{5179}{57600},+ 0, \tfrac{7571}{16695}, \tfrac{393}{640}, -\tfrac{92097}{339200}, \tfrac{187}{2100},+ \tfrac{1}{40})$. The order-5 row is the advancing formula. This is SciPy''s RK45+ tableau; the seventh stage is the FSAL row.'+- params:+ method: dormand-prince+ coefficient: a2_1+ number: 1/5+ param-latex: $a_{21}$+- params:+ method: dormand-prince+ coefficient: c3+ number: 3/10+ param-latex: $c_3$+- params:+ method: dormand-prince+ coefficient: a3_1+ number: 3/40+ param-latex: $a_{31}$+- params:+ method: dormand-prince+ coefficient: a3_2+ number: 9/40+ param-latex: $a_{32}$+- params:+ method: dormand-prince+ coefficient: c4+ number: 4/5+ param-latex: $c_4$+- params:+ method: dormand-prince+ coefficient: a4_1+ number: 44/45+ param-latex: $a_{41}$+- params:+ method: dormand-prince+ coefficient: a4_2+ number: -56/15+ param-latex: $a_{42}$+- params:+ method: dormand-prince+ coefficient: a4_3+ number: 32/9+ param-latex: $a_{43}$+- params:+ method: dormand-prince+ coefficient: c5+ number: 8/9+ param-latex: $c_5$+- params:+ method: dormand-prince+ coefficient: a5_1+ number: 19372/6561+ param-latex: $a_{51}$+- params:+ method: dormand-prince+ coefficient: a5_2+ number: -25360/2187+ param-latex: $a_{52}$+- params:+ method: dormand-prince+ coefficient: a5_3+ number: 64448/6561+ param-latex: $a_{53}$+- params:+ method: dormand-prince+ coefficient: a5_4+ number: -212/729+ param-latex: $a_{54}$+- params:+ method: dormand-prince+ coefficient: c6+ number: '1'+ param-latex: $c_6$+- params:+ method: dormand-prince+ coefficient: a6_1+ number: 9017/3168+ param-latex: $a_{61}$+- params:+ method: dormand-prince+ coefficient: a6_2+ number: -355/33+ param-latex: $a_{62}$+- params:+ method: dormand-prince+ coefficient: a6_3+ number: 46732/5247+ param-latex: $a_{63}$+- params:+ method: dormand-prince+ coefficient: a6_4+ number: 49/176+ param-latex: $a_{64}$+- params:+ method: dormand-prince+ coefficient: a6_5+ number: -5103/18656+ param-latex: $a_{65}$+- params:+ method: dormand-prince+ coefficient: c7+ number: '1'+ param-latex: $c_7$+- params:+ method: dormand-prince+ coefficient: a7_1+ number: 35/384+ param-latex: $a_{71}$+- params:+ method: dormand-prince+ coefficient: a7_3+ number: 500/1113+ param-latex: $a_{73}$+- params:+ method: dormand-prince+ coefficient: a7_4+ number: 125/192+ param-latex: $a_{74}$+- params:+ method: dormand-prince+ coefficient: a7_5+ number: -2187/6784+ param-latex: $a_{75}$+- params:+ method: dormand-prince+ coefficient: a7_6+ number: 11/84+ param-latex: $a_{76}$+- params:+ method: dormand-prince+ coefficient: b5_1+ number: 35/384+ param-latex: $b^{(5)}_1$+- params:+ method: dormand-prince+ coefficient: b5_3+ number: 500/1113+ param-latex: $b^{(5)}_3$+- params:+ method: dormand-prince+ coefficient: b5_4+ number: 125/192+ param-latex: $b^{(5)}_4$+- params:+ method: dormand-prince+ coefficient: b5_5+ number: -2187/6784+ param-latex: $b^{(5)}_5$+- params:+ method: dormand-prince+ coefficient: b5_6+ number: 11/84+ param-latex: $b^{(5)}_6$+- params:+ method: dormand-prince+ coefficient: b4_1+ number: 5179/57600+ param-latex: $b^{(4)}_1$+- params:+ method: dormand-prince+ coefficient: b4_3+ number: 7571/16695+ param-latex: $b^{(4)}_3$+- params:+ method: dormand-prince+ coefficient: b4_4+ number: 393/640+ param-latex: $b^{(4)}_4$+- params:+ method: dormand-prince+ coefficient: b4_5+ number: -92097/339200+ param-latex: $b^{(4)}_5$+- params:+ method: dormand-prince+ coefficient: b4_6+ number: 187/2100+ param-latex: $b^{(4)}_6$+- params:+ method: dormand-prince+ coefficient: b4_7+ number: 1/40+ param-latex: $b^{(4)}_7$
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