51.3.8 problem 8

Internal problem ID [10351]
Book : First order enumerated odes
Section : section 3. First order odes solved using Laplace method
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 07:22:30 PM
CAS classification : [_linear]

\begin{align*} t y^{\prime }+y&=\sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (1\right )&=0 \\ \end{align*}

✗ Maple

ode:=t*diff(y(t),t)+y(t) = sin(t); 
ic:=[y(1) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 19

ode=t*D[y[t],t]+y[t]==Sin[t]; 
ic=y[1]==0; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {\int _1^t\sin (K[1])dK[1]}{t} \end{align*}

✓ Sympy. Time used: 0.152 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), t) + y(t) - sin(t),0) 
ics = {y(1): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {- \cos {\left (t \right )} + \cos {\left (1 \right )}}{t} \]