15.20.2 problem 2
Internal
problem
ID
[3310]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
38,
page
173
Problem
number
:
2
Date
solved
:
Monday, January 27, 2025 at 07:33:07 AM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} x&=y-{y^{\prime }}^{3} \end{align*}
✓ Solution by Maple
Time used: 0.034 (sec). Leaf size: 211
dsolve(x=y(x)-diff(y(x),x)^3,y(x), singsol=all)
\begin{align*}
x -\frac {3 \left (-x +y \left (x \right )\right )^{{2}/{3}}}{2}-3 \left (-x +y \left (x \right )\right )^{{1}/{3}}-3 \ln \left (\left (-x +y \left (x \right )\right )^{{1}/{3}}-1\right )-c_{1} &= 0 \\
x +\frac {3 \left (-x +y \left (x \right )\right )^{{2}/{3}}}{4}-\frac {3 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{{2}/{3}}}{4}+\frac {3 \left (-x +y \left (x \right )\right )^{{1}/{3}}}{2}+\frac {3 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{{1}/{3}}}{2}+6 \ln \left (2\right )-3 \ln \left (-2 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{{1}/{3}}-2 \left (-x +y \left (x \right )\right )^{{1}/{3}}-4\right )-c_{1} &= 0 \\
x +\frac {3 \left (-x +y \left (x \right )\right )^{{2}/{3}}}{4}+\frac {3 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{{2}/{3}}}{4}+\frac {3 \left (-x +y \left (x \right )\right )^{{1}/{3}}}{2}-\frac {3 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{{1}/{3}}}{2}+6 \ln \left (2\right )-3 \ln \left (2 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{{1}/{3}}-2 \left (-x +y \left (x \right )\right )^{{1}/{3}}-4\right )-c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 9.004 (sec). Leaf size: 271
DSolve[x==y[x]-D[y[x],x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {3}{2} (y(x)-x)^{2/3}+3 \sqrt [3]{y(x)-x}+3 \log \left (\sqrt [3]{y(x)-x}-1\right )-x&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {1}{4} i \left (-12 i \arctan \left (\frac {2 \sqrt [3]{y(x)-x}+1}{\sqrt {3}}\right )-3 i \left (\sqrt {3}-i\right ) (y(x)-x)^{2/3}+6 i \left (\sqrt {3}+i\right ) \sqrt [3]{y(x)-x}+6 \log \left ((y(x)-x)^{2/3}+\sqrt [3]{y(x)-x}+1\right )-4 x\right )&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {y(x)}{2}+\frac {1}{4} \left (2 (x-y(x))+\frac {3}{2} \left (1-i \sqrt {3}\right ) (y(x)-x)^{2/3}+3 \left (1+i \sqrt {3}\right ) \sqrt [3]{y(x)-x}-6 \log \left (2 i \sqrt [3]{y(x)-x}+\sqrt {2+2 i \sqrt {3}}\right )\right )&=c_1,y(x)\right ] \\
\end{align*}