30.2.21 problem 21

Internal problem ID [7449]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.3, Linear equations. Exercises. page 54
Problem number : 21
Date solved : Tuesday, September 30, 2025 at 04:35:41 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=-\frac {15 \sqrt {2}\, \pi ^{2}}{32} \\ \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 16

ode:=cos(x)*diff(y(x),x)+y(x)*sin(x) = 2*x*cos(x)^2; 
ic:=[y(1/4*Pi) = -15/32*2^(1/2)*Pi^2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 17

ode=Cos[x]*D[y[x],x]+y[x]*Sin[x]==2*x*Cos[x]^2; 
ic={y[Pi/4]==-15*Sqrt[2]*Pi^2/32}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (x^2-\pi ^2\right ) \cos (x) \end{align*}

✓ Sympy. Time used: 0.351 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*cos(x)**2 + y(x)*sin(x) + cos(x)*Derivative(y(x), x),0) 
ics = {y(pi/4): -15*sqrt(2)*pi**2/32} 
dsolve(ode,func=y(x),ics=ics)
 

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