29.11.17 problem 308

Internal problem ID [4908]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 11
Problem number : 308
Date solved : Monday, January 27, 2025 at 09:47:35 AM
CAS classification : [_rational, _Riccati]

\begin{align*} \left (a^{2}+x^{2}\right ) y^{\prime }&=a^{2}+3 y x -2 y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 217

dsolve((a^2+x^2)*diff(y(x),x) = a^2+3*x*y(x)-2*y(x)^2,y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {2 a \left (c_{1} \left (i a -x \right ) a^{2} \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, \operatorname {HeunCPrime}\left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {-i a +x}{i a +x}\right )+\left (i a -x \right ) a^{2} \sqrt {\frac {i x +a}{a}}\, \operatorname {HeunCPrime}\left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {-i a +x}{i a +x}\right )+\frac {c_{1} \sqrt {2}\, \left (i a x -\frac {1}{2} a^{2}+\frac {1}{2} x^{2}\right ) x \sqrt {\frac {i x -a}{a}}}{2}-\frac {\sqrt {\frac {i x +a}{a}}\, \left (i a^{3}-3 i x^{2} a +3 x \,a^{2}-x^{3}\right )}{4}\right )}{\left (i \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, c_{1} x +\frac {\sqrt {\frac {i x +a}{a}}\, \left (i x -a \right )}{2}\right ) \left (i a +x \right )^{2}} \]

✓ Solution by Mathematica

Time used: 1.231 (sec). Leaf size: 63

DSolve[(a^2+x^2)D[y[x],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*}