11.17 problem 308

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: 0.032 (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)
 

\[ y \left (x \right ) = -\frac {a \left (-\frac {\left (i x -a \right ) \sqrt {2}\, c_{1} \sqrt {\frac {i x -a}{a}}\, \operatorname {HeunC}\left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )}{2}+\left (i x -a \right ) \sqrt {\frac {i x +a}{a}}\, \operatorname {HeunC}\left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )+\left (i x +a \right ) \left (-c_{1} \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, \operatorname {HeunCPrime}\left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )+\sqrt {\frac {i x +a}{a}}\, \operatorname {HeunCPrime}\left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )\right )\right )}{\left (i a +x \right ) \left (c_{1} \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, \operatorname {HeunC}\left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )-\sqrt {\frac {i x +a}{a}}\, \operatorname {HeunC}\left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {i x +a}{i x -a}\right )\right )} \]

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*}