Internal problem ID [3484]
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`]]
\[ \boxed {{y^{\prime }}^{2}+x^{2}-4 y=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 141
dsolve(diff(y(x),x)^2+x^2 = 4*y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {x^{2}}{2}+\frac {{\mathrm e}^{2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{\frac {c_{1}}{2}}}{2}\right )+\ln \left (2\right )-c_{1}}}{4}+\frac {{\mathrm e}^{\operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{\frac {c_{1}}{2}}}{2}\right )+\frac {\ln \left (2\right )}{2}-\frac {c_{1}}{2}} x}{2} \\ y \left (x \right ) = \frac {x^{2} \left (2 \operatorname {LambertW}\left (-\frac {\sqrt {2}\, x c_{1}}{2}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {\sqrt {2}\, x c_{1}}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (-\frac {\sqrt {2}\, x c_{1}}{2}\right )^{2}} \\ y \left (x \right ) = \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_{1}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_{1}}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {\sqrt {2}\, x c_{1}}{2}\right )^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.797 (sec). Leaf size: 162
DSolve[(y'[x])^2+x^2==4 y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )+\frac {x \left (-\sqrt {4 y(x)-x^2}\right )+\left (x^2-2 y(x)\right ) \log \left (2 y(x)-x^2\right )+2 y(x)}{2 \left (x^2-2 y(x)\right )}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x \sqrt {4 y(x)-x^2}+\left (x^2-2 y(x)\right ) \log \left (2 y(x)-x^2\right )+2 y(x)}{2 \left (x^2-2 y(x)\right )}-\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )=c_1,y(x)\right ] \\ \end{align*}