4.38.44 $y(x)y^{″}(x)=a$

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

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

Book solution method
TO DO

Mathematica ✓
cpu = 0.209895 (sec), leaf count = 111

$$\{\{y(x)\to \exp (-\frac{c_1+2a{\text{erf}}^{-1}(-i\sqrt{\frac{2}{\pi }}\sqrt{a{e}^{\frac{c_1}{a}}(c_2+x){}^2}){}^2}{2a})\},\{y(x)\to \exp (-\frac{c_1+2a{\text{erf}}^{-1}\left(i\sqrt{\frac{2}{\pi }}\sqrt{a{e}^{\frac{c_1}{a}}(c_2+x){}^2}\right){}^2}{2a})\}\}$$

Maple ✓
cpu = 0.389 (sec), leaf count = 54

$$\{{\int }^{y\left(x\right)}\frac{1}{\sqrt{2a\ln \left(\mathit{\_}\mathit{a}\right)-2\mathit{\_}\mathit{C}\mathit{1}a}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-\frac{1}{\sqrt{-2a(\mathit{\_}\mathit{C}\mathit{1}-\ln \left(\mathit{\_}\mathit{a}\right))}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

DSolve[y[x]*y''[x] == a,y[x],x]

Mathematica raw output

{{y[x] -> E^(-(C[1] + 2*a*InverseErf[(-I)*Sqrt[2/Pi]*Sqrt[a*E^(C[1]/a)*(x + C[2]
)^2]]^2)/(2*a))}, {y[x] -> E^(-(C[1] + 2*a*InverseErf[I*Sqrt[2/Pi]*Sqrt[a*E^(C[1
]/a)*(x + C[2])^2]]^2)/(2*a))}}

Maple raw input

dsolve(y(x)*diff(diff(y(x),x),x) = a, y(x),'implicit')

Maple raw output

Intat(1/(2*a*ln(_a)-2*_C1*a)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(-2*a*(_C1-ln(
_a)))^(1/2),_a = y(x))-x-_C2 = 0