29.36.18 problem 1086
Internal
problem
ID
[5645]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
36
Problem
number
:
1086
Date
solved
:
Monday, January 27, 2025 at 01:01:15 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{4}&=\left (y-a \right )^{3} \left (y-b \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.105 (sec). Leaf size: 131
dsolve(diff(y(x),x)^4 = (y(x)-a)^3*(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 )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{4}}}d \textit {\_a} -c_{1} &= 0 \\
x -i \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{4}}}d \textit {\_a} \right )-c_{1} &= 0 \\
x +i \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{4}}}d \textit {\_a} \right )-c_{1} &= 0 \\
x +\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{4}}}d \textit {\_a} -c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.577 (sec). Leaf size: 333
DSolve[(D[y[x],x])^4 == (y[x]-a)^3 (y[x]-b)^2 ,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [-\sqrt [4]{-1} x+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\sqrt [4]{-1} x+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [-(-1)^{3/4} x+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [(-1)^{3/4} x+c_1\right ] \\
y(x)\to a \\
y(x)\to b \\
\end{align*}