23.1.423 problem 413
Internal
problem
ID
[5030]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
413
Date
solved
:
Tuesday, September 30, 2025 at 11:27:37 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime } \left (a +\cos \left (\frac {x}{2}\right )^{2}\right )&=y \tan \left (\frac {x}{2}\right ) \left (1+a +\cos \left (\frac {x}{2}\right )^{2}-y\right ) \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 254
ode:=diff(y(x),x)*(a+cos(1/2*x)^2) = y(x)*tan(1/2*x)*(1+a+cos(1/2*x)^2-y(x));
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (a +\cos \left (\frac {x}{2}\right )^{2}\right )^{\frac {1}{a}} \left (a +1\right ) \cos \left (\frac {x}{2}\right )^{-\frac {2}{a}}}{\sin \left (\frac {x}{2}\right )^{2} 4^{\frac {1}{a}} \left ({\mathrm e}^{i x}+1\right )^{-\frac {2}{a}} \left (a \,{\mathrm e}^{i x}+\frac {{\mathrm e}^{2 i x}}{4}+\frac {{\mathrm e}^{i x}}{2}+\frac {1}{4}\right )^{\frac {1}{a}} {\mathrm e}^{\frac {i \pi \left (2 \,\operatorname {csgn}\left (i \cos \left (\frac {x}{2}\right )\right )-2 \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )\right )-2 \,\operatorname {csgn}\left (i {\mathrm e}^{-\frac {i x}{2}}\right )+2 \,\operatorname {csgn}\left (i \cos \left (\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\frac {i x}{2}}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )\right )+2 \,\operatorname {csgn}\left (i {\mathrm e}^{i x}\right )-2 \,\operatorname {csgn}\left (i {\mathrm e}^{\frac {i x}{2}}\right )-\operatorname {csgn}\left (i \left (2 a +1+\cos \left (x \right )\right )\right )+\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right )+\operatorname {csgn}\left (i \left (4 a \,{\mathrm e}^{i x}+2 \,{\mathrm e}^{i x}+{\mathrm e}^{2 i x}+1\right )\right )-\operatorname {csgn}\left (i \left (2 a +1+\cos \left (x \right )\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left (4 a \,{\mathrm e}^{i x}+2 \,{\mathrm e}^{i x}+{\mathrm e}^{2 i x}+1\right )\right )\right )}{2 a}}+c_1 \cos \left (\frac {x}{2}\right )^{2} \left (a +1\right )}
\]
✓ Mathematica. Time used: 1.656 (sec). Leaf size: 196
ode=D[y[x],x]*(a+Cos[x/2]^2)==y[x]*Tan[x/2]*(1+a+Cos[x/2]^2-y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\cos ^{-\frac {2 (a+1)}{a}}\left (\frac {x}{2}\right ) \left (a+\cos ^2\left (\frac {x}{2}\right )\right )^{\frac {1}{a}}}{-\int _1^x-\frac {2 \cos ^{-3-\frac {2}{a}}\left (\frac {K[1]}{2}\right ) \left (\cos ^2\left (\frac {K[1]}{2}\right )+a\right )^{\frac {1}{a}} \sin \left (\frac {K[1]}{2}\right )}{2 a+\cos (K[1])+1}dK[1]+c_1}\\ y(x)&\to 0\\ y(x)&\to -\frac {\cos ^{-\frac {2 (a+1)}{a}}\left (\frac {x}{2}\right ) \left (a+\cos ^2\left (\frac {x}{2}\right )\right )^{\frac {1}{a}}}{\int _1^x-\frac {2 \cos ^{-3-\frac {2}{a}}\left (\frac {K[1]}{2}\right ) \left (\cos ^2\left (\frac {K[1]}{2}\right )+a\right )^{\frac {1}{a}} \sin \left (\frac {K[1]}{2}\right )}{2 a+\cos (K[1])+1}dK[1]} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq((a + cos(x/2)**2)*Derivative(y(x), x) - (a - y(x) + cos(x/2)**2 + 1)*y(x)*tan(x/2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out