Internal problem ID [3671]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 945.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}-4 a^{2} x y^{\prime }+y a^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.18 (sec). Leaf size: 121
dsolve(y(x)*diff(y(x),x)^2-4*a^2*x*diff(y(x),x)+a^2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a}^{2}+2 a^{2}+\sqrt {-\textit {\_a}^{2} a^{2}+4 a^{4}}}{\textit {\_a} \left (\textit {\_a}^{2}-3 a^{2}\right )}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}-2 a^{2}+\sqrt {-\textit {\_a}^{2} a^{2}+4 a^{4}}}{\textit {\_a} \left (\textit {\_a}^{2}-3 a^{2}\right )}d \textit {\_a} \right )+c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.492 (sec). Leaf size: 758
DSolve[y[x] (y'[x])^2-4 a^2 x y'[x]+a^2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \left (\sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )-8 \text {ArcTan}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )+4 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )-2 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )\right )+8 \left (4 a^2-\frac {y(x)^2}{x^2}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-2 a}}{2 \sqrt {a}}\right )}{6 \sqrt {a} \sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {\frac {y(x)}{a x}+2} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}=-\log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \left (\sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )+8 \text {ArcTan}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )-4 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )+2 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )\right )-8 \left (4 a^2-\frac {y(x)^2}{x^2}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-2 a}}{2 \sqrt {a}}\right )}{6 \sqrt {a} \sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {\frac {y(x)}{a x}+2} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}=-\log (x)+c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}