29.21.14 problem 590

Internal problem ID [5184]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 590
Date solved : Tuesday, January 28, 2025 at 02:41:32 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} x^{7} y y^{\prime }&=2 x^{2}+2+5 x^{3} y \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 120

dsolve(x^7*y(x)*diff(y(x),x) = 2*x^2+2+5*x^3*y(x),y(x), singsol=all)
 

\[ -\frac {\left (\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x^{3} y \left (x \right )+1\right )^{2}}{x^{2}}\right ) y \left (x \right ) x^{3}-c_{1} x +\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x^{3} y \left (x \right )+1\right )^{2}}{x^{2}}\right )\right ) \left (\frac {x^{6} y \left (x \right )^{2}+2 x^{3} y \left (x \right )+x^{2}+1}{x^{2}}\right )^{{1}/{4}}+2 x^{2}}{\left (\frac {x^{6} y \left (x \right )^{2}+2 x^{3} y \left (x \right )+x^{2}+1}{x^{2}}\right )^{{1}/{4}} x} = 0 \]

✓ Solution by Mathematica

Time used: 0.383 (sec). Leaf size: 98

DSolve[x^7 y[x] D[y[x],x]==2(1+x^2)+5 x^3 y[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [c_1=\frac {\frac {i \left (x^3 y(x)+1\right ) \sqrt [4]{x^4 y(x)^2+\frac {1}{x^2}+2 x y(x)+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},-\frac {\left (y(x) x^3+1\right )^2}{x^2}\right )}{2 x}+i x}{\sqrt [4]{-\frac {\left (x^3 y(x)+1\right )^2}{x^2}-1}},y(x)\right ] \]