60.1.194 problem 197
Internal
problem
ID
[10208]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
197
Date
solved
:
Sunday, March 30, 2025 at 03:30:51 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} \cos \left (x \right ) y^{\prime }-y^{4}-y \sin \left (x \right )&=0 \end{align*}
✓ Maple. Time used: 0.015 (sec). Leaf size: 179
ode:=cos(x)*diff(y(x),x)-y(x)^4-y(x)*sin(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (\cos \left (x \right )^{3} \left (-\cos \left (x \right )^{3} c_1 +2 \cos \left (x \right )^{2} \sin \left (x \right )+\sin \left (x \right )\right )^{2}\right )^{{1}/{3}} \sec \left (x \right )}{\cos \left (x \right )^{3} c_1 -2 \cos \left (x \right )^{2} \sin \left (x \right )-\sin \left (x \right )} \\
y &= \frac {\left (\cos \left (x \right )^{3} \left (-\cos \left (x \right )^{3} c_1 +2 \cos \left (x \right )^{2} \sin \left (x \right )+\sin \left (x \right )\right )^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) \sec \left (x \right )}{-2 \cos \left (x \right )^{3} c_1 +4 \cos \left (x \right )^{2} \sin \left (x \right )+2 \sin \left (x \right )} \\
y &= \frac {\left (-1+i \sqrt {3}\right ) \left (\cos \left (x \right )^{3} \left (-\cos \left (x \right )^{3} c_1 +2 \cos \left (x \right )^{2} \sin \left (x \right )+\sin \left (x \right )\right )^{2}\right )^{{1}/{3}} \sec \left (x \right )}{2 \cos \left (x \right )^{3} c_1 -4 \cos \left (x \right )^{2} \sin \left (x \right )-2 \sin \left (x \right )} \\
\end{align*}
✓ Mathematica. Time used: 1.104 (sec). Leaf size: 109
ode=Cos[x]*D[y[x],x] - y[x]^4 - y[x]*Sin[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\
y(x)\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\
y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{-\sin ^3(x)+c_1 \cos ^3(x)-3 \sin (x) \cos ^2(x)}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 7.904 (sec). Leaf size: 104
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x)**4 - y(x)*sin(x) + cos(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}}{2}, \ y{\left (x \right )} = \sqrt [3]{\frac {1}{C_{1} \cos ^{3}{\left (x \right )} - 2 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \sin {\left (x \right )}}}\right ]
\]