60.1.396 problem 398
Internal
problem
ID
[10410]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
398
Date
solved
:
Monday, January 27, 2025 at 07:40:36 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} {y^{\prime }}^{2}-3 x y^{{2}/{3}} y^{\prime }+9 y^{{5}/{3}}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.221 (sec). Leaf size: 150
dsolve(diff(y(x),x)^2-3*x*y(x)^(2/3)*diff(y(x),x)+9*y(x)^(5/3) = 0,y(x), singsol=all)
\begin{align*}
y &= \frac {x^{6}}{64} \\
y &= 0 \\
\ln \left (x \right )+\frac {\ln \left (\frac {64 y}{x^{6}}-1\right )}{6}-\frac {\ln \left (4 \left (\frac {y}{x^{6}}\right )^{{1}/{3}}-1\right )}{6}-\frac {\ln \left (16 \left (\frac {y}{x^{6}}\right )^{{2}/{3}}+4 \left (\frac {y}{x^{6}}\right )^{{1}/{3}}+1\right )}{6}+\frac {\ln \left (\frac {y}{x^{6}}\right )}{6}-\frac {\sqrt {-\frac {y \left (\frac {y}{x^{6}}\right )^{{1}/{3}} \left (4 \left (\frac {y}{x^{6}}\right )^{{1}/{3}}-1\right )}{x^{6}}}\, \operatorname {arctanh}\left (\sqrt {-4 \left (\frac {y}{x^{6}}\right )^{{1}/{3}}+1}\right )}{\left (\frac {y}{x^{6}}\right )^{{2}/{3}} \sqrt {-4 \left (\frac {y}{x^{6}}\right )^{{1}/{3}}+1}}-c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 9.334 (sec). Leaf size: 418
DSolve[9*y[x]^(5/3) - 3*x*y[x]^(2/3)*D[y[x],x] + D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {\sqrt {x^2-4 \sqrt [3]{y(x)}} \sqrt {x^2 y(x)^{4/3}-4 y(x)^{5/3}}}{8 y(x)-2 x^2 y(x)^{2/3}}+\frac {\sqrt {\left (x^2-4 \sqrt [3]{y(x)}\right ) y(x)^{4/3}} \log \left (\sqrt {x^2-4 \sqrt [3]{y(x)}}-x\right )}{\sqrt {x^2-4 \sqrt [3]{y(x)}} y(x)^{2/3}}+\log \left (4 y(x)^{4/3}-x^2 y(x)\right )-\log \left (x^2 \left (-y(x)^{2/3}\right )+\sqrt {x^2-4 \sqrt [3]{y(x)}} \sqrt {x^2 y(x)^{4/3}-4 y(x)^{5/3}}+4 y(x)\right )&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {1}{6} \left (\frac {3 \sqrt {x^2 y(x)^{4/3}-4 y(x)^{5/3}}}{\sqrt {x^2-4 \sqrt [3]{y(x)}} y(x)^{2/3}}+6 \log \left (4 y(x)^{4/3}-x^2 y(x)\right )-6 \log \left (x^2 y(x)^{2/3}+\sqrt {x^2-4 \sqrt [3]{y(x)}} \sqrt {x^2 y(x)^{4/3}-4 y(x)^{5/3}}-4 y(x)\right )\right )-\frac {\sqrt {\left (x^2-4 \sqrt [3]{y(x)}\right ) y(x)^{4/3}} \log \left (\sqrt {x^2-4 \sqrt [3]{y(x)}}-x\right )}{\sqrt {x^2-4 \sqrt [3]{y(x)}} y(x)^{2/3}}&=c_1,y(x)\right ] \\
y(x)\to 0 \\
\end{align*}