7.5.56 problem 56

Internal problem ID [160]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 56
Date solved : Friday, February 07, 2025 at 07:59:06 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+p \left (x \right ) y&=q \left (x \right ) y^{n} \end{align*}

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 55

dsolve(diff(y(x),x)+p(x)*y(x)=q(x)*y(x)^n,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\int p \left (x \right )d x} {\left (-n \left (\int q \left (x \right ) {\mathrm e}^{-\left (\int p \left (x \right )d x \right ) \left (n -1\right )}d x \right )+c_1 +\int q \left (x \right ) {\mathrm e}^{-\left (\int p \left (x \right )d x \right ) \left (n -1\right )}d x \right )}^{-\frac {1}{n -1}} \]

✓ Solution by Mathematica

Time used: 11.748 (sec). Leaf size: 71

DSolve[D[y[x],x]+p[x]*y[x]==q[x]*y[x]^n,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (\exp \left (-\left ((n-1) \int _1^x-p(K[1])dK[1]\right )\right ) \left (-(n-1) \int _1^x\exp \left ((n-1) \int _1^{K[2]}-p(K[1])dK[1]\right ) q(K[2])dK[2]+c_1\right )\right ){}^{\frac {1}{1-n}} \]