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 : Saturday, March 29, 2025 at 04:37:27 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 55

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

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

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

ode=D[y[x],x]+p[x]*y[x]==q[x]*y[x]^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 3.922 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(p(x)*y(x) - q(x)*y(x)**n + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\left (C_{1} - n \int q{\left (x \right )} e^{- \left (n - 1\right ) \int p{\left (x \right )}\, dx}\, dx + \int q{\left (x \right )} e^{- \left (n - 1\right ) \int p{\left (x \right )}\, dx}\, dx\right ) e^{\left (n - 1\right ) \int p{\left (x \right )}\, dx}\right )^{- \frac {1}{n - 1}} \]