29.26.29 problem 765
Internal
problem
ID
[5350]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
26
Problem
number
:
765
Date
solved
:
Monday, January 27, 2025 at 11:16:52 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.118 (sec). Leaf size: 220
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b) = 0,y(x), singsol=all)
\begin{align*}
\frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \left (-\ln \left (2\right )+\ln \left (-a -b +2 y \left (x \right )+2 \sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\right )\right )}{\sqrt {y \left (x \right )-a}\, \sqrt {y \left (x \right )-b}}-\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-a}\, \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \left (-\ln \left (2\right )+\ln \left (-a -b +2 y \left (x \right )+2 \sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\right )\right )}{\sqrt {y \left (x \right )-a}\, \sqrt {y \left (x \right )-b}}+\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-a}\, \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.224 (sec). Leaf size: 89
DSolve[(D[y[x],x])^2+ f[x]*(y[x]-a)*(y[x]-b)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{2} \left ((b-a) \cosh \left (\int _1^x-i \sqrt {f(K[2])}dK[2]+c_1\right )+a+b\right ) \\
y(x)\to \frac {1}{2} \left ((b-a) \cosh \left (\int _1^xi \sqrt {f(K[3])}dK[3]+c_1\right )+a+b\right ) \\
y(x)\to a \\
y(x)\to b \\
\end{align*}