40.3.6 problem 23 (j)
Internal
problem
ID
[6610]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
5.
Equations
of
first
order
and
first
degree
(Exact
equations).
Supplemetary
problems.
Page
33
Problem
number
:
23
(j)
Date
solved
:
Monday, January 27, 2025 at 02:15:52 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 2 u^{2}+2 u v+\left (u^{2}+v^{2}\right ) v^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 317
dsolve(2*(u^2+u*v(u))+(u^2+v(u)^2)*diff(v(u),u)=0,v(u), singsol=all)
\begin{align*}
v &= -\frac {2 \left (u^{2} c_1 -\frac {\left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_1}\, \left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
v &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}}}{4 \sqrt {c_1}}-\frac {\sqrt {c_1}\, u^{2} \left (i \sqrt {3}-1\right )}{\left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
v &= \frac {4 i \sqrt {3}\, c_1 \,u^{2}+i \left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}} \sqrt {3}+4 u^{2} c_1 -\left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}}}{4 \left (4-8 u^{3} c_1^{{3}/{2}}+4 \sqrt {8 u^{6} c_1^{3}-4 u^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 21.166 (sec). Leaf size: 593
DSolve[2*(u^2+u*v[u])+(u^2+v[u]^2)*D[ v[u],u]==0,v[u],u,IncludeSingularSolutions -> True]
\begin{align*}
v(u)\to \frac {\sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} u^2}{\sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}} \\
v(u)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) u^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}} \\
v(u)\to \frac {\left (1-i \sqrt {3}\right ) u^2}{2^{2/3} \sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-2 u^3+\sqrt {8 u^6-4 e^{3 c_1} u^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\
v(u)\to \sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}-\frac {u^2}{\sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}} \\
v(u)\to \frac {\left (1-i \sqrt {3}\right ) u^2+\left (-1-i \sqrt {3}\right ) \left (\sqrt {2} \sqrt {u^6}-u^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}} \\
v(u)\to \frac {\left (1+i \sqrt {3}\right ) u^2+i \left (\sqrt {3}+i\right ) \left (\sqrt {2} \sqrt {u^6}-u^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {2} \sqrt {u^6}-u^3}} \\
\end{align*}