86.3.11 problem 9

Internal problem ID [23102]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 4. Linear equations of the first order. Exercise 4a at page 56
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:21:53 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 15

ode:=diff(y(x),x)-y(x)*sin(x) = sin(x); 
ic:=[y(0) = 2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -1+3 \,{\mathrm e}^{-\cos \left (x \right )} {\mathrm e} \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 17

ode=D[y[x],x]-y[x]*Sin[x]==Sin[x]; 
ic={y[0]==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 3 e^{1-\cos (x)}-1 \end{align*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)*sin(x) - sin(x) + Derivative(y(x), x),0) 
ics = {y(0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = -1 + 3 e e^{- \cos {\left (x \right )}} \]