Internal problem ID [3458]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 720.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {f \relax (x ) y^{m} y^{\prime }+g \relax (x ) y^{m +1}+h \relax (x ) y^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 221
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)
\[ y \relax (x ) = \left (-m \left (\int \frac {{\mathrm e}^{\int -\frac {g \relax (x ) n}{f \relax (x )}d x} {\mathrm e}^{\left (\int \frac {g \relax (x )}{f \relax (x )}d x \right ) m} {\mathrm e}^{\int \frac {g \relax (x )}{f \relax (x )}d x} h \relax (x )}{f \relax (x )}d x \right )+n \left (\int \frac {{\mathrm e}^{\int -\frac {g \relax (x ) n}{f \relax (x )}d x} {\mathrm e}^{\left (\int \frac {g \relax (x )}{f \relax (x )}d x \right ) m} {\mathrm e}^{\int \frac {g \relax (x )}{f \relax (x )}d x} h \relax (x )}{f \relax (x )}d x \right )+c_{1}-\left (\int \frac {{\mathrm e}^{\int -\frac {g \relax (x ) n}{f \relax (x )}d x} {\mathrm e}^{\left (\int \frac {g \relax (x )}{f \relax (x )}d x \right ) m} {\mathrm e}^{\int \frac {g \relax (x )}{f \relax (x )}d x} h \relax (x )}{f \relax (x )}d x \right )\right )^{\frac {1}{-n +m +1}} {\mathrm e}^{\int -\frac {m g \relax (x )}{\left (-n +m +1\right ) f \relax (x )}d x} {\mathrm e}^{\frac {\left (\int \frac {g \relax (x )}{f \relax (x )}d x \right ) n}{-n +m +1}} {\mathrm e}^{\int -\frac {g \relax (x )}{f \relax (x ) \left (-n +m +1\right )}d x} \]
✓ Solution by Mathematica
Time used: 43.425 (sec). Leaf size: 185
DSolve[f[x] y[x]^m y'[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 ((-m+n-1) \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\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 ((-m+n-1) \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right ) h(K[2])}{f(K[2])}dK[2]\right ){}^{\frac {1}{m-n+1}} \\ \end{align*}