73.1.25 problem 5

Internal problem ID [19798]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 3. Solutions of first-order equations. Exercises at page 47
Problem number : 5
Date solved : Thursday, October 02, 2025 at 04:43:53 PM
CAS classification : [_linear]

\begin{align*} x^{\prime } t +x g \left (t \right )&=h \left (t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 35

ode:=t*diff(x(t),t)+x(t)*g(t) = h(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \left (\int \frac {h \left (t \right ) {\mathrm e}^{\int \frac {g \left (t \right )}{t}d t}}{t}d t +c_1 \right ) {\mathrm e}^{-\int \frac {g \left (t \right )}{t}d t} \]

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 63

ode=t*D[x[t],t]+x[t]*g[t]==h[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \exp \left (\int _1^t-\frac {g(K[1])}{K[1]}dK[1]\right ) \left (\int _1^t\frac {\exp \left (-\int _1^{K[2]}-\frac {g(K[1])}{K[1]}dK[1]\right ) h(K[2])}{K[2]}dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 29.889 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
g = Function("g") 
h = Function("h") 
ode = Eq(t*Derivative(x(t), t) + g(t)*x(t) - h(t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ \left (e^{\int \frac {g{\left (t \right )}}{t}\, dt} - \int \frac {g{\left (t \right )} e^{\int \frac {g{\left (t \right )}}{t}\, dt}}{t}\, dt\right ) x{\left (t \right )} + \int \frac {\left (g{\left (t \right )} x{\left (t \right )} - h{\left (t \right )}\right ) e^{\int \frac {g{\left (t \right )}}{t}\, dt}}{t}\, dt = C_{1} \]