29.34.18 problem 1020
Internal
problem
ID
[5589]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1020
Date
solved
:
Monday, January 27, 2025 at 12:12:55 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 210
dsolve(diff(y(x),x)^3+f(x)*(y(x)-a)^2*(y(x)-b)^2 = 0,y(x), singsol=all)
\begin{align*}
\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right )\right )^{{2}/{3}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{2} \left (y \left (x \right )-b \right )^{2}\right )^{{1}/{3}}d \textit {\_a}}{\left (\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )\right )^{{2}/{3}}}+c_{1} &= 0 \\
\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right )\right )^{{2}/{3}}}d \textit {\_a} +\frac {\left (1+i \sqrt {3}\right ) \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{2} \left (y \left (x \right )-b \right )^{2}\right )^{{1}/{3}}d \textit {\_a} \right )}{2 \left (\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )\right )^{{2}/{3}}}+c_{1} &= 0 \\
\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right )\right )^{{2}/{3}}}d \textit {\_a} -\frac {\left (i \sqrt {3}-1\right ) \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{2} \left (y \left (x \right )-b \right )^{2}\right )^{{1}/{3}}d \textit {\_a} \right )}{2 \left (\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )\right )^{{2}/{3}}}+c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.817 (sec). Leaf size: 287
DSolve[(D[y[x],x])^3 +f[x] (y[x]-a)^2 (y[x]-b)^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-\sqrt [3]{f(K[1])}dK[1]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-(-1)^{2/3} \sqrt [3]{f(K[3])}dK[3]+c_1\right ] \\
y(x)\to a \\
y(x)\to b \\
\end{align*}