68.4.21 problem 21
Internal
problem
ID
[17201]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Exercises
2.2,
page
39
Problem
number
:
21
Date
solved
:
Thursday, October 02, 2025 at 01:52:19 PM
CAS
classification
:
[_separable]
\begin{align*} \left (2-\frac {5}{y^{2}}\right ) y^{\prime }+4 \cos \left (x \right )^{2}&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 107
ode:=(2-5/y(x)^2)*diff(y(x),x)+4*cos(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -c_1 -\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}-\frac {\sqrt {-158+16 \left (2 c_1 +x \right ) \sin \left (2 x \right )+16 x^{2}+64 c_1 x +64 c_1^{2}-2 \cos \left (4 x \right )}}{8} \\
y &= -c_1 -\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}+\frac {\sqrt {-158+16 \left (2 c_1 +x \right ) \sin \left (2 x \right )+16 x^{2}+64 c_1 x +64 c_1^{2}-2 \cos \left (4 x \right )}}{8} \\
\end{align*}
✓ Mathematica. Time used: 0.614 (sec). Leaf size: 200
ode=(2-5/y[x]^2)*D[y[x],x]+4*Cos[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{4} \left (\int _1^x-4 \cos ^2(K[1])dK[1]-\sqrt {-40+\left (\int _1^x-4 \cos ^2(K[1])dK[1]+c_1\right ){}^2}+c_1\right )\\ y(x)&\to \frac {1}{4} \left (\int _1^x-4 \cos ^2(K[1])dK[1]+\sqrt {-40+\left (\int _1^x-4 \cos ^2(K[1])dK[1]+c_1\right ){}^2}+c_1\right )\\ y(x)&\to 0\\ y(x)&\to \frac {1}{4} \left (\int _1^x-4 \cos ^2(K[1])dK[1]-\sqrt {\int _1^x-4 \cos ^2(K[1])dK[1]{}^2-40}\right )\\ y(x)&\to \frac {1}{4} \left (\int _1^x-4 \cos ^2(K[1])dK[1]+\sqrt {\int _1^x-4 \cos ^2(K[1])dK[1]{}^2-40}\right ) \end{align*}
✓ Sympy. Time used: 1.683 (sec). Leaf size: 122
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((2 - 5/y(x)**2)*Derivative(y(x), x) + 4*cos(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {C_{1}}{4} - \frac {x}{2} - \frac {\sqrt {C_{1}^{2} - 4 C_{1} x - 2 C_{1} \sin {\left (2 x \right )} + 4 x^{2} + 4 x \sin {\left (2 x \right )} - \frac {\cos {\left (4 x \right )}}{2} - \frac {79}{2}}}{4} - \frac {\sin {\left (2 x \right )}}{4}, \ y{\left (x \right )} = \frac {C_{1}}{4} - \frac {x}{2} + \frac {\sqrt {C_{1}^{2} - 4 C_{1} x - 2 C_{1} \sin {\left (2 x \right )} + 4 x^{2} + 4 x \sin {\left (2 x \right )} - \frac {\cos {\left (4 x \right )}}{2} - \frac {79}{2}}}{4} - \frac {\sin {\left (2 x \right )}}{4}\right ]
\]