29.34.17 problem 1019
Internal
problem
ID
[5588]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1019
Date
solved
:
Monday, January 27, 2025 at 12:12:54 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}&=\left (y-a \right )^{2} \left (y-b \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.051 (sec). Leaf size: 142
dsolve(diff(y(x),x)^3 = (y(x)-a)^2*(y(x)-b)^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= a \\
y \left (x \right ) &= b \\
x -\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{3}}}d \textit {\_a} -c_{1} &= 0 \\
\frac {2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{3}}}d \textit {\_a} \right )+i \left (-c_{1} +x \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\
\frac {-2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{3}}}d \textit {\_a} \right )+i \left (-c_{1} +x \right ) \sqrt {3}-x +c_{1}}{i \sqrt {3}-1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.120 (sec). Leaf size: 246
DSolve[(D[y[x],x])^3 ==(y[x]-a)^2 (y[x]-b)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ][x+c_1] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [-\sqrt [3]{-1} x+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [(-1)^{2/3} x+c_1\right ] \\
y(x)\to a \\
y(x)\to b \\
\end{align*}