Internal problem ID [10199]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
26. Equations solvable for \(x\). Page 55
Problem number: Ex 2.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {a^{2} \left (y^{\prime }\right )^{2} y-2 y^{\prime } x +y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 65
dsolve(a^2*y(x)*diff(y(x),x)^2-2*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {x}{a} \\ y \relax (x ) = \frac {x}{a} \\ y \relax (x ) = 0 \\ y \relax (x ) = {\mathrm e}^{\RootOf \left ({\mathrm e}^{2 \textit {\_Z}} \left (\tanh ^{2}\left (-\textit {\_Z} +c_{1}-\ln \relax (x )\right )\right ) a^{2}-\left (\tanh ^{2}\left (-\textit {\_Z} +c_{1}-\ln \relax (x )\right )\right )+1\right )} x \\ \end{align*}
✓ Solution by Mathematica
Time used: 12.556 (sec). Leaf size: 180
DSolve[a^2*y[x]*(y'[x])^2-2*x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{\frac {a^2 c_1}{2}} \sqrt {e^{a^2 c_1}-8 i x}}{4 a} \\ y(x)\to \frac {e^{\frac {a^2 c_1}{2}} \sqrt {e^{a^2 c_1}-8 i x}}{4 a} \\ y(x)\to -\frac {e^{\frac {a^2 c_1}{2}} \sqrt {e^{a^2 c_1}+8 i x}}{4 a} \\ y(x)\to \frac {e^{\frac {a^2 c_1}{2}} \sqrt {e^{a^2 c_1}+8 i x}}{4 a} \\ y(x)\to -\frac {x}{a} \\ y(x)\to \frac {x}{a} \\ \end{align*}