77.1.86 problem 113 (page 172)
Internal
problem
ID
[17976]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
113
(page
172)
Date
solved
:
Tuesday, January 28, 2025 at 11:19:14 AM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{2}&=y^{2} \ln \left (y\right ) \end{align*}
✓ Solution by Maple
Time used: 0.118 (sec). Leaf size: 73
dsolve(y(x)*diff(y(x),x$2)+diff(y(x),x)^2=y(x)^2*ln(y(x)),y(x), singsol=all)
\begin{align*}
-4 \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {8 \textit {\_a}^{4} \ln \left (\textit {\_a} \right )-2 \textit {\_a}^{4}+16 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
4 \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {8 \textit {\_a}^{4} \ln \left (\textit {\_a} \right )-2 \textit {\_a}^{4}+16 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.627 (sec). Leaf size: 311
DSolve[y[x]*D[y[x],{x,2}]+D[y[x],x]^2==y[x]^2*Log[y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {2 \sqrt {2} K[1]}{\sqrt {4 \log (K[1]) K[1]^4-K[1]^4+8 c_1}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2 \sqrt {2} K[2]}{\sqrt {4 \log (K[2]) K[2]^4-K[2]^4+8 c_1}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {2 \sqrt {2} K[1]}{\sqrt {4 \log (K[1]) K[1]^4-K[1]^4+8 (-1) c_1}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {2 \sqrt {2} K[1]}{\sqrt {4 \log (K[1]) K[1]^4-K[1]^4+8 c_1}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2 \sqrt {2} K[2]}{\sqrt {4 \log (K[2]) K[2]^4-K[2]^4+8 (-1) c_1}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2 \sqrt {2} K[2]}{\sqrt {4 \log (K[2]) K[2]^4-K[2]^4+8 c_1}}dK[2]\&\right ][x+c_2] \\
\end{align*}