29.4.7 problem 95

Internal problem ID [4698]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 4
Problem number : 95
Date solved : Monday, January 27, 2025 at 09:32:11 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.080 (sec). Leaf size: 51

dsolve(diff(y(x),x) = f(x)*y(x)+g(x)*y(x)^k,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 11.650 (sec). Leaf size: 67

DSolve[D[y[x],x]==f[x] y[x]+g[x]y[x]^k,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (\exp \left (-\left ((k-1) \int _1^xf(K[1])dK[1]\right )\right ) \left (-(k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]+c_1\right )\right ){}^{\frac {1}{1-k}} \]