29.27.3 problem 768
Internal
problem
ID
[5353]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
27
Problem
number
:
768
Date
solved
:
Monday, January 27, 2025 at 11:16:58 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 )^{2} \left (y-b \right ) \left (y-c \right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 342
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)^2*(y(x)-b)*(y(x)-c) = 0,y(x), singsol=all)
\begin{align*}
-\frac {\ln \left (\frac {-2 \sqrt {\left (y \left (x \right )-b \right ) \left (y \left (x \right )-c \right )}\, \sqrt {\left (a -b \right ) \left (-c +a \right )}+\left (-2 a +b +c \right ) y \left (x \right )+\left (b +c \right ) a -2 b c}{-y \left (x \right )+a}\right ) \sqrt {a^{2}-a b -a c +b c}\, \sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-c}}{\left (-c +a \right ) \left (a -b \right ) \sqrt {-c y \left (x \right )+b c +y \left (x \right )^{2}-b y \left (x \right )}}+\frac {\int _{}^{x}\sqrt {f \left (\textit {\_a} \right ) \left (-y \left (x \right )+b \right ) \left (y \left (x \right )-c \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-c}\, \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
-\frac {\ln \left (\frac {-2 \sqrt {\left (y \left (x \right )-b \right ) \left (y \left (x \right )-c \right )}\, \sqrt {\left (a -b \right ) \left (-c +a \right )}+\left (-2 a +b +c \right ) y \left (x \right )+\left (b +c \right ) a -2 b c}{-y \left (x \right )+a}\right ) \sqrt {a^{2}-a b -a c +b c}\, \sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-c}}{\left (-c +a \right ) \left (a -b \right ) \sqrt {-c y \left (x \right )+b c +y \left (x \right )^{2}-b y \left (x \right )}}-\frac {\int _{}^{x}\sqrt {f \left (\textit {\_a} \right ) \left (-y \left (x \right )+b \right ) \left (y \left (x \right )-c \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-c}\, \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.748 (sec). Leaf size: 251
DSolve[(D[y[x],x])^2+f[x]*(y[x]-a)^2*(y[x]-b)*(y[x]-c)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {c (a-b)+b (a-c) \tan ^2\left (\frac {1}{2} \sqrt {b-a} \sqrt {a-c} \left (\int _1^x-i \sqrt {f(K[1])}dK[1]+c_1\right )\right )}{(a-c) \tan ^2\left (\frac {1}{2} \sqrt {b-a} \sqrt {a-c} \left (\int _1^x-i \sqrt {f(K[1])}dK[1]+c_1\right )\right )+a-b} \\
y(x)\to \frac {c (a-b)+b (a-c) \tan ^2\left (\frac {1}{2} \sqrt {b-a} \sqrt {a-c} \left (\int _1^xi \sqrt {f(K[2])}dK[2]+c_1\right )\right )}{(a-c) \tan ^2\left (\frac {1}{2} \sqrt {b-a} \sqrt {a-c} \left (\int _1^xi \sqrt {f(K[2])}dK[2]+c_1\right )\right )+a-b} \\
\end{align*}