38.6.35 problem 35

Internal problem ID [8465]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.3 Linear equations. Exercises 2.3 at page 63
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 05:37:34 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }-\sin \left (x \right ) y&=2 \sin \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 13

ode:=diff(y(x),x)-y(x)*sin(x) = 2*sin(x); 
ic:=[y(1/2*Pi) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -2+3 \,{\mathrm e}^{-\cos \left (x \right )} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 59

ode=D[y[x],x]-Sin[x]*y[x]==2*Sin[x]; 
ic={y[Pi/2]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (\int _{\frac {\pi }{2}}^x\sin (K[1])dK[1]\right ) \left (\int _{\frac {\pi }{2}}^x2 \exp \left (-\int _{\frac {\pi }{2}}^{K[2]}\sin (K[1])dK[1]\right ) \sin (K[2])dK[2]+1\right ) \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)*sin(x) - 2*sin(x) + Derivative(y(x), x),0) 
ics = {y(pi/2): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = -2 + 3 e^{- \cos {\left (x \right )}} \]