29.36.23 problem 1092
Internal
problem
ID
[5650]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
36
Problem
number
:
1092
Date
solved
:
Monday, January 27, 2025 at 01:01:26 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} {y^{\prime }}^{4}-4 x^{2} y {y^{\prime }}^{2}+16 x y^{2} y^{\prime }-16 y^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.129 (sec). Leaf size: 120
dsolve(diff(y(x),x)^4-4*x^2*y(x)*diff(y(x),x)^2+16*x*y(x)^2*diff(y(x),x)-16*y(x)^3 = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {x^{4}}{16} \\
y \left (x \right ) &= 0 \\
y \left (x \right ) \left (\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}-x \right )^{-\frac {2 \sqrt {x^{2} y \left (x \right )-4 y \left (x \right )^{{3}/{2}}}}{\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\, \sqrt {y \left (x \right )}}} \left (x +\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\right )^{\frac {2 \sqrt {x^{2} y \left (x \right )-4 y \left (x \right )^{{3}/{2}}}}{\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\, \sqrt {y \left (x \right )}}}-c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 7.695 (sec). Leaf size: 779
DSolve[(D[y[x],x])^4 -4 x^2 y[x] (D[y[x],x])^2+16 x y[x]^2 D[y[x],x]-16 y[x]^3==0,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*}