Internal problem ID [3710]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 986.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {\left (\left (-4 a^{2}+1\right ) x^{2}+y^{2}\right ) \left (y^{\prime }\right )^{2}-8 a^{2} x y y^{\prime }+x^{2}+\left (-4 a^{2}+1\right ) y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.578 (sec). Leaf size: 173
dsolve(((-4*a^2+1)*x^2+y(x)^2)*diff(y(x),x)^2-8*a^2*x*y(x)*diff(y(x),x)+x^2+(-4*a^2+1)*y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a}^{3}+8 \textit {\_a} \,a^{2}+\sqrt {4 \textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+8 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+4 a^{2}-1}-\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-8 \textit {\_a} \,a^{2}+\sqrt {4 \textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+8 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+4 a^{2}-1}+\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} \right )+c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.623 (sec). Leaf size: 321
DSolve[((1-4 a^2)x^2+y[x]^2) (y'[x])^2 - 8 a^2 x y[x] y'[x]+x^2+(1-4 a^2)y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 \sqrt {2 a-1} \sqrt {2 a+1}+8 \text {$\#$1} a^2-\text {$\#$1}+\sqrt {2 a-1} \sqrt {2 a+1}\&,\frac {-\text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+4 a^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-\log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{-3 \text {$\#$1}^2+2 \text {$\#$1} \sqrt {2 a-1} \sqrt {2 a+1}+8 a^2-1}\&\right ]=-\log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 \sqrt {2 a-1} \sqrt {2 a+1}-8 \text {$\#$1} a^2+\text {$\#$1}+\sqrt {2 a-1} \sqrt {2 a+1}\&,\frac {-\text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+4 a^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-\log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{-3 \text {$\#$1}^2-2 \text {$\#$1} \sqrt {2 a-1} \sqrt {2 a+1}+8 a^2-1}\&\right ]=-\log (x)+c_1,y(x)\right ] \\ \end{align*}