64.4.12 problem 12
Internal
problem
ID
[13230]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
2,
section
2.2
(Separable
equations).
Exercises
page
47
Problem
number
:
12
Date
solved
:
Monday, March 31, 2025 at 07:41:56 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \left (2 s^{2}+2 s t +t^{2}\right ) s^{\prime }+s^{2}+2 s t -t^{2}&=0 \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 344
ode:=(2*s(t)^2+2*s(t)*t+t^2)*diff(s(t),t)+s(t)^2+2*s(t)*t-t^2 = 0;
dsolve(ode,s(t), singsol=all);
\begin{align*}
s &= \frac {\left (4 t^{3} c_1^{3}+2+\sqrt {17 t^{6} c_1^{6}+16 t^{3} c_1^{3}+4}\right )^{{1}/{3}}-\frac {t^{2} c_1^{2}}{\left (4 t^{3} c_1^{3}+2+\sqrt {17 t^{6} c_1^{6}+16 t^{3} c_1^{3}+4}\right )^{{1}/{3}}}-t c_1}{2 c_1} \\
s &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 t^{3} c_1^{3}+2+\sqrt {17 t^{6} c_1^{6}+16 t^{3} c_1^{3}+4}\right )^{{2}/{3}}+c_1 t \left (2 \left (4 t^{3} c_1^{3}+2+\sqrt {17 t^{6} c_1^{6}+16 t^{3} c_1^{3}+4}\right )^{{1}/{3}}+c_1 t \left (i \sqrt {3}-1\right )\right )}{4 \left (4 t^{3} c_1^{3}+2+\sqrt {17 t^{6} c_1^{6}+16 t^{3} c_1^{3}+4}\right )^{{1}/{3}} c_1} \\
s &= \frac {\left (i \sqrt {3}-1\right ) \left (4 t^{3} c_1^{3}+2+\sqrt {17 t^{6} c_1^{6}+16 t^{3} c_1^{3}+4}\right )^{{2}/{3}}+c_1 \left (-2 \left (4 t^{3} c_1^{3}+2+\sqrt {17 t^{6} c_1^{6}+16 t^{3} c_1^{3}+4}\right )^{{1}/{3}}+c_1 \left (1+i \sqrt {3}\right ) t \right ) t}{4 \left (4 t^{3} c_1^{3}+2+\sqrt {17 t^{6} c_1^{6}+16 t^{3} c_1^{3}+4}\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 43.459 (sec). Leaf size: 616
ode=(2*s[t]^2+2*s[t]*t+t^2)*D[s[t],t]+(s[t]^2+2*s[t]*t-t^2)==0;
ic={};
DSolve[{ode,ic},s[t],t,IncludeSingularSolutions->True]
\begin{align*}
s(t)\to \frac {1}{2} \left (\sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}-\frac {t^2}{\sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}}-t\right ) \\
s(t)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}+\frac {2 \left (1+i \sqrt {3}\right ) t^2}{\sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}}-4 t\right ) \\
s(t)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}+\frac {2 \left (1-i \sqrt {3}\right ) t^2}{\sqrt [3]{4 t^3+\sqrt {17 t^6+16 e^{3 c_1} t^3+4 e^{6 c_1}}+2 e^{3 c_1}}}-4 t\right ) \\
s(t)\to \frac {1}{2} \left (\sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}-\frac {t^2}{\sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}}-t\right ) \\
s(t)\to \frac {1}{4} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}+\frac {\left (1-i \sqrt {3}\right ) t^2}{\sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}}-2 t\right ) \\
s(t)\to \frac {1}{4} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}+\frac {\left (1+i \sqrt {3}\right ) t^2}{\sqrt [3]{\sqrt {17} \sqrt {t^6}+4 t^3}}-2 t\right ) \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
s = Function("s")
ode = Eq(-t**2 + 2*t*s(t) + (t**2 + 2*t*s(t) + 2*s(t)**2)*Derivative(s(t), t) + s(t)**2,0)
ics = {}
dsolve(ode,func=s(t),ics=ics)
Timed Out