75.7.6 problem 180
Internal
problem
ID
[16728]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
7,
Total
differential
equations.
The
integrating
factor.
Exercises
page
61
Problem
number
:
180
Date
solved
:
Monday, March 31, 2025 at 03:12:17 PM
CAS
classification
:
[_exact]
\begin{align*} \frac {\sin \left (2 x \right )}{y}+x +\left (y-\frac {\sin \left (x \right )^{2}}{y^{2}}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 397
ode:=sin(2*x)/y(x)+x+(y(x)-sin(x)^2/y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_1 \,x^{4}+144 x^{2} c_1^{2}+96 c_1^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{{2}/{3}}-12 x^{2}-24 c_1}{6 \left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_1 \,x^{4}+144 x^{2} c_1^{2}+96 c_1^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_1 \,x^{4}+144 x^{2} c_1^{2}+96 c_1^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{{2}/{3}}+\left (x^{2}+2 c_1 \right ) \left (i \sqrt {3}-1\right )}{\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_1 \,x^{4}+144 x^{2} c_1^{2}+96 c_1^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_1 \,x^{4}+144 x^{2} c_1^{2}+96 c_1^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{{2}/{3}} \left (i \sqrt {3}-1\right )}{12}+\left (x^{2}+2 c_1 \right ) \left (1+i \sqrt {3}\right )}{\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_1 \,x^{4}+144 x^{2} c_1^{2}+96 c_1^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 0.353 (sec). Leaf size: 74
ode=( Sin[2*x]/y[x]+x )+( y[x]-Sin[x]^2/y[x]^2 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x\left (2 K[1]+\frac {2 \sin (2 K[1])}{y(x)}\right )dK[1]+\int _1^{y(x)}\left (\frac {\cos (2 x)-1}{K[2]^2}+2 K[2]-\int _1^x-\frac {2 \sin (2 K[1])}{K[2]^2}dK[1]\right )dK[2]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x + (y(x) - sin(x)**2/y(x)**2)*Derivative(y(x), x) + sin(2*x)/y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out