Internal problem ID [3485]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 752.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+x^{2}-4 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.266 (sec). Leaf size: 141
dsolve(diff(y(x),x)^2+x^2 = 4*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x^{2}}{2}+\frac {{\mathrm e}^{2 \LambertW \left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )+\ln \relax (2)+c_{1}}}{4}+\frac {{\mathrm e}^{\LambertW \left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )+\frac {\ln \relax (2)}{2}+\frac {c_{1}}{2}} x}{2} \\ y \relax (x ) = \frac {x^{2} \left (2 \LambertW \left (-\frac {\sqrt {2}\, x c_{1}}{2}\right )^{2}+2 \LambertW \left (-\frac {\sqrt {2}\, x c_{1}}{2}\right )+1\right )}{4 \LambertW \left (-\frac {\sqrt {2}\, x c_{1}}{2}\right )^{2}} \\ y \relax (x ) = \frac {x^{2} \left (2 \LambertW \left (\frac {\sqrt {2}\, x c_{1}}{2}\right )^{2}+2 \LambertW \left (\frac {\sqrt {2}\, x c_{1}}{2}\right )+1\right )}{4 \LambertW \left (\frac {\sqrt {2}\, x c_{1}}{2}\right )^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.735 (sec). Leaf size: 192
DSolve[(y'[x])^2+x^2==4 y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {2 y(x)-x \sqrt {4 y(x)-x^2}}{2 x^2-4 y(x)}-\log \left (\sqrt {4 y(x)-x^2}-i x\right )+\log \left (x \sqrt {4 y(x)-x^2}-i x^2+(2+2 i) y(x)\right )=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x \sqrt {4 y(x)-x^2}+2 y(x)}{2 x^2-4 y(x)}-\log \left (\sqrt {4 y(x)-x^2}-i x\right )+\log \left (x \sqrt {4 y(x)-x^2}-i x^2-(2-2 i) y(x)\right )=c_1,y(x)\right ] \\ \end{align*}