29.27.2 problem 767
Internal
problem
ID
[5352]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
27
Problem
number
:
767
Date
solved
:
Monday, January 27, 2025 at 11:16:55 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 ) \left (y-c \right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.271 (sec). Leaf size: 156
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b)*(y(x)-c) = 0,y(x), singsol=all)
\begin{align*}
\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )}d \textit {\_a}}{\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-c \right )}}+c_{1} &= 0 \\
\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} +\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )}d \textit {\_a}}{\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-c \right )}}+c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 39.505 (sec). Leaf size: 228
DSolve[(D[y[x],x])^2+f[x]*(y[x]-a)(y[x]-b)*(y[x]-c)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x-\sqrt {f(K[1])}dK[1]\right )|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x-\sqrt {f(K[1])}dK[1]\right )|\frac {a-c}{a-b}\right ){}^2-a+b\right ) \\
y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x\sqrt {f(K[2])}dK[2]\right )|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x\sqrt {f(K[2])}dK[2]\right )|\frac {a-c}{a-b}\right ){}^2-a+b\right ) \\
y(x)\to a \\
y(x)\to b \\
y(x)\to c \\
\end{align*}