Internal problem ID [3056]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 11
Problem number: 308.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {\left (a^{2}+x^{2}\right ) y^{\prime }-a^{2}-3 y x +2 y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 280
dsolve((a^2+x^2)*diff(y(x),x) = a^2+3*x*y(x)-2*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-\frac {c_{1} \left (i x -a \right ) \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, \HeunC \left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )}{2}+\left (i x -a \right ) \sqrt {\frac {i x +a}{a}}\, \HeunC \left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )+\left (-c_{1} \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, \HeunCPrime \left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )+\HeunCPrime \left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right ) \sqrt {\frac {i x +a}{a}}\right ) \left (i x +a \right )\right ) a}{\left (c_{1} \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, \HeunC \left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )-\sqrt {\frac {i x +a}{a}}\, \HeunC \left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )\right ) \left (i a +x \right )} \]
✓ Solution by Mathematica
Time used: 1.067 (sec). Leaf size: 63
DSolve[(a^2+x^2)y'[x]==a^2+3 x y[x]-2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a^2 c_1 (-x) \sqrt {a^2+x^2}+a^2+2 x^2}{2 x-a^2 c_1 \sqrt {a^2+x^2}} \\ y(x)\to x \\ \end{align*}