y′′(x)2 = a + by(x)

4.42.12 $y^{″} (x)^{2} = a + b y (x)$

ODE
$y^{″} (x)^{2} = a + b y (x)$ ODE Classiﬁcation

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.850654 (sec), leaf count = 119

${Solve [\frac{(a + b y (x))^{2}_{2} F_{1} (\frac{1}{2}, \frac{2}{3}; \frac{5}{3}; - \frac{4 (a + b y (x))^{3 / 2}}{3 b c_{1}})^{2}}{b^{2} c_{1}} = (c_{2} + x)^{2}, y (x)], Solve [\frac{(a + b y (x))^{2}_{2} F_{1} (\frac{1}{2}, \frac{2}{3}; \frac{5}{3}; \frac{4 (a + b y (x))^{3 / 2}}{3 b c_{1}})^{2}}{b^{2} c_{1}} = (c_{2} + x)^{2}, y (x)]}$

Maple ✓
cpu = 0.441 (sec), leaf count = 173

${\int^{y (x)} b \sqrt{3} \frac{1}{\sqrt{b (4_a \sqrt{b_a + a} b + 4 a \sqrt{b_a + a} -_C 1)}} d_a - x -_C 2 = 0, \int^{y (x)} - 3 \frac{b}{\sqrt{- 12 b ({(b_a + a)}^{3 / 2} -_C 1 / 4)}} d_a - x -_C 2 = 0, \int^{y (x)} 3 \frac{b}{\sqrt{- 12 b ({(b_a + a)}^{3 / 2} -_C 1 / 4)}} d_a - x -_C 2 = 0, \int^{y (x)} - b \sqrt{3} \frac{1}{\sqrt{b (4_a \sqrt{b_a + a} b + 4 a \sqrt{b_a + a} -_C 1)}} d_a - x -_C 2 = 0, y (x) = - \frac{a}{b}}$ Mathematica raw input

DSolve[y''[x]^2 == a + b*y[x],y[x],x]

Mathematica raw output

{Solve[(Hypergeometric2F1[1/2, 2/3, 5/3, (-4*(a + b*y[x])^(3/2))/(3*b*C[1])]^2*(
a + b*y[x])^2)/(b^2*C[1]) == (x + C[2])^2, y[x]], Solve[(Hypergeometric2F1[1/2,
2/3, 5/3, (4*(a + b*y[x])^(3/2))/(3*b*C[1])]^2*(a + b*y[x])^2)/(b^2*C[1]) == (x
+ C[2])^2, y[x]]}

Maple raw input

dsolve(diff(diff(y(x),x),x)^2 = a+b*y(x), y(x),'implicit')

Maple raw output

y(x) = -a/b, Intat(b*3^(1/2)/(b*(4*_a*(_a*b+a)^(1/2)*b+4*a*(_a*b+a)^(1/2)-_C1))^
(1/2),_a = y(x))-x-_C2 = 0, Intat(-b*3^(1/2)/(b*(4*_a*(_a*b+a)^(1/2)*b+4*a*(_a*b
+a)^(1/2)-_C1))^(1/2),_a = y(x))-x-_C2 = 0, Intat(-3*b/(-12*b*((_a*b+a)^(3/2)-1/
4*_C1))^(1/2),_a = y(x))-x-_C2 = 0, Intat(3*b/(-12*b*((_a*b+a)^(3/2)-1/4*_C1))^(
1/2),_a = y(x))-x-_C2 = 0

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4.42.12 y″(x)2=a+by(x)

4.42.12 $y^{″} (x)^{2} = a + b y (x)$