29.8.15 problem 220

Internal problem ID [4820]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 8
Problem number : 220
Date solved : Monday, January 27, 2025 at 09:41:25 AM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} x y^{\prime }&=y f \left (x^{m} y^{n}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.122 (sec). Leaf size: 39

dsolve(x*diff(y(x),x) = y(x)*f(x^m*y(x)^n),y(x), singsol=all)
 

\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {1}{\left (f \left (x^{m} \textit {\_a}^{n}\right ) n +m \right ) \textit {\_a}}d \textit {\_a} -\frac {\ln \left (x \right )}{n}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.374 (sec). Leaf size: 186

DSolve[x D[y[x],x]==y[x] f[x^m y[x]^n] ,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {n}{\left (m+n f\left (x^m K[2]^n\right )\right ) K[2]}-\int _1^x\left (\frac {n^2 K[1]^{m-1} K[2]^{n-1} f''\left (K[1]^m K[2]^n\right )}{m+n f\left (K[1]^m K[2]^n\right )}-\frac {n^3 f\left (K[1]^m K[2]^n\right ) K[1]^{m-1} K[2]^{n-1} f''\left (K[1]^m K[2]^n\right )}{\left (m+n f\left (K[1]^m K[2]^n\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {n f\left (K[1]^m y(x)^n\right )}{\left (m+n f\left (K[1]^m y(x)^n\right )\right ) K[1]}dK[1]=c_1,y(x)\right ] \]