77.1.32 problem 49 (page 56)

Internal problem ID [17922]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 49 (page 56)
Date solved : Tuesday, January 28, 2025 at 11:12:46 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 115

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

\begin{align*} y &= \frac {c_{1} a^{2} b^{2} x +x +\sqrt {\left (a^{2}+b^{2}\right ) \left (-1+\left (a^{4}+a^{2} x^{2}+b^{2} x^{2}\right ) c_{1} \right )}}{a^{4} c_{1} -1} \\ y &= \frac {c_{1} a^{2} b^{2} x +x -\sqrt {\left (a^{2}+b^{2}\right ) \left (-1+\left (a^{4}+a^{2} x^{2}+b^{2} x^{2}\right ) c_{1} \right )}}{a^{4} c_{1} -1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.936 (sec). Leaf size: 228

DSolve[(x*(x+y[x])+a^2)*D[y[x],x]==y[x]*(x+y[x])+b^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {a^2-\frac {1}{\frac {a^2}{a^4+a^2 x^2+b^2 x^2}-\frac {x}{\left (a^4+a^2 x^2+b^2 x^2\right )^{3/2} \sqrt {-\frac {1}{\left (a^2+b^2\right ) \left (a^4+a^2 x^2+b^2 x^2\right )}+c_1}}}+x^2}{x} \\ y(x)\to -\frac {a^2-\frac {1}{\frac {a^2}{a^4+a^2 x^2+b^2 x^2}+\frac {x}{\left (a^4+a^2 x^2+b^2 x^2\right )^{3/2} \sqrt {-\frac {1}{\left (a^2+b^2\right ) \left (a^4+a^2 x^2+b^2 x^2\right )}+c_1}}}+x^2}{x} \\ y(x)\to \frac {b^2 x}{a^2} \\ \end{align*}