77.1.75 problem 94 (page 135)
Internal
problem
ID
[17965]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
94
(page
135)
Date
solved
:
Tuesday, January 28, 2025 at 11:18:42 AM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} {y^{\prime }}^{4}&=4 y \left (x y^{\prime }-2 y\right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.151 (sec). Leaf size: 120
dsolve(diff(y(x),x)^4=4*y(x)*(x*diff(y(x),x)-2*y(x))^2,y(x), singsol=all)
\begin{align*}
y &= \frac {x^{4}}{16} \\
y &= 0 \\
y \left (\sqrt {x^{2}-4 \sqrt {y}}-x \right )^{-\frac {2 \sqrt {x^{2} y-4 y^{{3}/{2}}}}{\sqrt {x^{2}-4 \sqrt {y}}\, \sqrt {y}}} \left (\sqrt {x^{2}-4 \sqrt {y}}+x \right )^{\frac {2 \sqrt {x^{2} y-4 y^{{3}/{2}}}}{\sqrt {x^{2}-4 \sqrt {y}}\, \sqrt {y}}}-c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.015 (sec). Leaf size: 779
DSolve[D[y[x],x]^4==4*y[x]*(x*D[y[x],x]-2*y[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {x^2 y(x)-4 y(x)^{3/2}}}{8 y(x)-2 x^2 \sqrt {y(x)}}+\frac {\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2-4 \sqrt {y(x)}}-x\right )}{\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)}}+\log \left (4 y(x)^{3/2}-x^2 y(x)\right )-\log \left (x^2 \left (-\sqrt {y(x)}\right )+\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {x^2 y(x)-4 y(x)^{3/2}}+4 y(x)\right )&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2+4 \sqrt {y(x)}}-x\right )}{\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}}+\frac {1}{4} \left (-\frac {2 \sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}}{\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}}+4 \log \left (\sqrt {x^2+4 \sqrt {y(x)}} y(x)\right )-4 \log \left (\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}-\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}\right )\right )&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {1}{4} \left (\frac {2 \sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}}{\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}}+4 \log \left (\sqrt {x^2+4 \sqrt {y(x)}} y(x)\right )-4 \log \left (\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}+\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}\right )\right )-\frac {\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2+4 \sqrt {y(x)}}-x\right )}{\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {1}{4} \left (\frac {2 \sqrt {x^2 y(x)-4 y(x)^{3/2}}}{\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)}}+4 \log \left (4 y(x)^{3/2}-x^2 y(x)\right )-4 \log \left (x^2 \sqrt {y(x)}+\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {x^2 y(x)-4 y(x)^{3/2}}-4 y(x)\right )\right )-\frac {\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2-4 \sqrt {y(x)}}-x\right )}{\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)}}&=c_1,y(x)\right ] \\
y(x)\to 0 \\
y(x)\to \frac {x^4}{16} \\
\end{align*}