Internal problem ID [2331]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 37, page 171
Problem number: 18.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
\[ \boxed {2 x^{2} y+{y^{\prime }}^{2}-y^{\prime } x^{3}=0} \]
✓ Solution by Maple
Time used: 0.218 (sec). Leaf size: 21
dsolve(2*x^2*y(x)+diff(y(x),x)^2=diff(y(x),x)*x^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {x^{4}}{8} \\ y \left (x \right ) &= c_{1} \left (x^{2}-2 c_{1} \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.642 (sec). Leaf size: 216
DSolve[2*x^2*y[x]+y'[x]^2==y'[x]*x^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\sqrt {x^6-8 x^2 y(x)} \log \left (\sqrt {x^4-8 y(x)}+x^2\right )}{2 x \sqrt {x^4-8 y(x)}}-\frac {\sqrt {x^6-8 x^2 y(x)} \log (y(x))}{4 x \sqrt {x^4-8 y(x)}}+\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [-\frac {\sqrt {x^6-8 x^2 y(x)} \log \left (\sqrt {x^4-8 y(x)}+x^2\right )}{2 x \sqrt {x^4-8 y(x)}}+\frac {\sqrt {x^6-8 x^2 y(x)} \log (y(x))}{4 x \sqrt {x^4-8 y(x)}}+\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to \frac {x^4}{8} \\ \end{align*}