Internal problem ID [3564]
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]
\[ \boxed {\left (a^{2}+x^{2}\right ) y^{\prime }-3 y x +2 y^{2}=a^{2}} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 235
dsolve((a^2+x^2)*diff(y(x),x) = a^2+3*x*y(x)-2*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2 \left (\sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, c_{1} a^{2} \left (i a -x \right ) \operatorname {HeunCPrime}\left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {-i a +x}{i a +x}\right )+\sqrt {\frac {i x +a}{a}}\, a^{2} \left (i a -x \right ) \operatorname {HeunCPrime}\left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {-i a +x}{i a +x}\right )+\frac {\sqrt {2}\, x \left (i a x -\frac {1}{2} a^{2}+\frac {1}{2} x^{2}\right ) c_{1} \sqrt {\frac {i x -a}{a}}}{2}-\frac {\sqrt {\frac {i x +a}{a}}\, \left (i a^{3}-3 i a \,x^{2}+3 x \,a^{2}-x^{3}\right )}{4}\right ) a}{\left (i \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, c_{1} x +\frac {\sqrt {\frac {i x +a}{a}}\, \left (i x -a \right )}{2}\right ) \left (i a +x \right )^{2}} \]
✓ Solution by Mathematica
Time used: 1.077 (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*}