79.5.4 problem (e)

Internal problem ID [21096]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises IX at page 45
Problem number : (e)
Date solved : Thursday, October 02, 2025 at 07:08:13 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 51

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

\begin{align*} y(x)&\to \exp \left (\int _1^x-p(K[1])dK[1]\right ) \left (\int _1^x\exp \left (-\int _1^{K[2]}-p(K[1])dK[1]\right ) q(K[2])dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 2.352 (sec). Leaf size: 41

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

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