88.4.10 problem 10
Internal
problem
ID
[23974]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
2.
Differential
equations
of
first
order.
Exercise
at
page
33
Problem
number
:
10
Date
solved
:
Thursday, October 02, 2025 at 09:48:20 PM
CAS
classification
:
[_separable]
\begin{align*} y^{2} \sec \left (x \right )^{2} y^{\prime }+x&=0 \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 107
ode:=y(x)^2*sec(x)^2*diff(y(x),x)+x = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-6 x \sin \left (2 x \right )-6 x^{2}-3 \cos \left (2 x \right )+8 c_1 +3\right )^{{1}/{3}}}{2} \\
y &= -\frac {\left (-6 x \sin \left (2 x \right )-6 x^{2}-3 \cos \left (2 x \right )+8 c_1 +3\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\
y &= \frac {\left (-6 x \sin \left (2 x \right )-6 x^{2}-3 \cos \left (2 x \right )+8 c_1 +3\right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.276 (sec). Leaf size: 126
ode=y[x]^2*Sec[x]^2*D[y[x],{x,1}]+x==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {1}{2} \sqrt [3]{-3} \sqrt [3]{-\cos (2 x)-2 \left (x^2+x \sin (2 x)-4 c_1\right )}\\ y(x)&\to \frac {1}{2} \sqrt [3]{3} \sqrt [3]{-\cos (2 x)-2 \left (x^2+x \sin (2 x)-4 c_1\right )}\\ y(x)&\to \frac {1}{2} (-1)^{2/3} \sqrt [3]{3} \sqrt [3]{-\cos (2 x)-2 \left (x^2+x \sin (2 x)-4 c_1\right )} \end{align*}
✓ Sympy. Time used: 4.025 (sec). Leaf size: 175
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x + y(x)**2*sec(x)**2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1} - \frac {3 x^{2} \sin ^{2}{\left (x \right )}}{4} - \frac {3 x^{2} \cos ^{2}{\left (x \right )}}{4} - \frac {3 x \sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {3 \sin ^{2}{\left (x \right )}}{4}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1} - \frac {3 x^{2} \sin ^{2}{\left (x \right )}}{4} - \frac {3 x^{2} \cos ^{2}{\left (x \right )}}{4} - \frac {3 x \sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {3 \sin ^{2}{\left (x \right )}}{4}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1} - \frac {3 x^{2} \sin ^{2}{\left (x \right )}}{4} - \frac {3 x^{2} \cos ^{2}{\left (x \right )}}{4} - \frac {3 x \sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {3 \sin ^{2}{\left (x \right )}}{4}}\right ]
\]