29.7.11 problem 186

Internal problem ID [4786]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 186
Date solved : Monday, January 27, 2025 at 09:37:49 AM
CAS classification : [[_homogeneous, `class D`], _Riccati]

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

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 13

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

\[ y \left (x \right ) = \tanh \left (\int f \left (x \right )d x +c_{1} \right ) x \]

✓ Solution by Mathematica

Time used: 0.465 (sec). Leaf size: 65

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

\begin{align*} y(x)\to \frac {x-x \exp \left (2 \left (\int _1^x-f(K[1])dK[1]+c_1\right )\right )}{1+\exp \left (2 \left (\int _1^x-f(K[1])dK[1]+c_1\right )\right )} \\ y(x)\to -x \\ y(x)\to x \\ \end{align*}