20.4.23 problem Problem 39
Internal
problem
ID
[3658]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
1,
First-Order
Differential
Equations.
Section
1.8,
Change
of
Variables.
page
79
Problem
number
:
Problem
39
Date
solved
:
Sunday, March 30, 2025 at 02:02:15 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }+\frac {\tan \left (x \right ) y}{2}&=2 y^{3} \sin \left (x \right ) \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 56
ode:=diff(y(x),x)+1/2*tan(x)*y(x) = 2*y(x)^3*sin(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {\left (-2 \sin \left (x \right )^{2}+c_1 \right ) \cos \left (x \right )}}{-2 \sin \left (x \right )^{2}+c_1} \\
y &= \frac {\sqrt {\left (-2 \sin \left (x \right )^{2}+c_1 \right ) \cos \left (x \right )}}{-2 \sin \left (x \right )^{2}+c_1} \\
\end{align*}
✓ Mathematica. Time used: 5.488 (sec). Leaf size: 227
ode=D[y[x],x]+1/2*Tan(x)*y[x]==2*y[x]^3*Sin[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {e^{\left .\frac {1}{4}\right /\text {Tan}} \sqrt [4]{\text {Tan}}}{\sqrt {e^{\frac {\text {Tan} x^2}{2}} \left (-i \sqrt {2 \pi } \text {erf}\left (\frac {\text {Tan} x+i}{\sqrt {2} \sqrt {\text {Tan}}}\right )+\sqrt {2 \pi } \text {erfi}\left (\frac {1+i \text {Tan} x}{\sqrt {2} \sqrt {\text {Tan}}}\right )+c_1 e^{\left .\frac {1}{2}\right /\text {Tan}} \sqrt {\text {Tan}}\right )}} \\
y(x)\to \frac {e^{\left .\frac {1}{4}\right /\text {Tan}} \sqrt [4]{\text {Tan}}}{\sqrt {e^{\frac {\text {Tan} x^2}{2}} \left (-i \sqrt {2 \pi } \text {erf}\left (\frac {\text {Tan} x+i}{\sqrt {2} \sqrt {\text {Tan}}}\right )+\sqrt {2 \pi } \text {erfi}\left (\frac {1+i \text {Tan} x}{\sqrt {2} \sqrt {\text {Tan}}}\right )+c_1 e^{\left .\frac {1}{2}\right /\text {Tan}} \sqrt {\text {Tan}}\right )}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 1.293 (sec). Leaf size: 32
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*y(x)**3*sin(x) + y(x)*tan(x)/2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {\frac {\cos {\left (x \right )}}{C_{1} + \cos {\left (2 x \right )}}}, \ y{\left (x \right )} = \sqrt {\frac {\cos {\left (x \right )}}{C_{1} + \cos {\left (2 x \right )}}}\right ]
\]