29.25.23 problem 720

Internal problem ID [5310]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 720
Date solved : Tuesday, January 28, 2025 at 02:41:45 PM
CAS classification : [_Bernoulli]

\begin{align*} f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{m +1}+h \left (x \right ) y^{n}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.118 (sec). Leaf size: 61

dsolve(f(x)*y(x)^m*diff(y(x),x)+g(x)*y(x)^(m+1)+h(x)*y(x)^n = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{-\int \frac {g \left (x \right )}{f \left (x \right )}d x} {\left (\left (n -m -1\right ) \left (\int \frac {h \left (x \right ) {\mathrm e}^{\left (-n +m +1\right ) \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )}}{f \left (x \right )}d x \right )+c_{1} \right )}^{\frac {1}{-n +m +1}} \]

✓ Solution by Mathematica

Time used: 12.490 (sec). Leaf size: 187

DSolve[f[x] y[x]^m D[y[x],x]+ g[x] y[x]^(m+1)+ h[x] y[x]^n==0,y[x],x,IncludeSingularSolutions -> True]
 

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