29.27.4 problem 770
Internal
problem
ID
[5354]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
27
Problem
number
:
770
Date
solved
:
Monday, January 27, 2025 at 11:17:00 AM
CAS
classification
:
[_separable]
\begin{align*} {y^{\prime }}^{2}&=f \left (x \right )^{2} \left (y-a \right ) \left (y-b \right ) \left (y-c \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 1.607 (sec). Leaf size: 284
dsolve(diff(y(x),x)^2 = f(x)^2*(y(x)-a)*(y(x)-b)*(y(x)-c)^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {c \,{\mathrm e}^{2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (-c +a \right ) \left (b -c \right )}}+\left (\left (4 b -2 c \right ) a -2 b c \right ) {\mathrm e}^{\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (-c +a \right ) \left (b -c \right )}}+c \left (a -b \right )^{2}}{{\mathrm e}^{2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (-c +a \right ) \left (b -c \right )}}+\left (2 a +2 b -4 c \right ) {\mathrm e}^{\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (-c +a \right ) \left (b -c \right )}}+a^{2}-2 a b +b^{2}} \\
y \left (x \right ) &= \frac {\left (\left (4 b -2 c \right ) a -2 b c \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (-c +a \right ) \left (b -c \right )}}+c \left ({\mathrm e}^{-2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (-c +a \right ) \left (b -c \right )}}+\left (a -b \right )^{2}\right )}{\left (2 a +2 b -4 c \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (-c +a \right ) \left (b -c \right )}}+a^{2}-2 a b +b^{2}+{\mathrm e}^{-2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (-c +a \right ) \left (b -c \right )}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.720 (sec). Leaf size: 223
DSolve[(D[y[x],x])^2==f[x]^2 *(y[x]-a)*(y[x]-b)*(y[x]-c)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {b (a-c)+a (b-c) \tan ^2\left (\frac {1}{2} \sqrt {c-a} \sqrt {b-c} \left (\int _1^x-f(K[1])dK[1]+c_1\right )\right )}{(b-c) \tan ^2\left (\frac {1}{2} \sqrt {c-a} \sqrt {b-c} \left (\int _1^x-f(K[1])dK[1]+c_1\right )\right )+a-c} \\
y(x)\to \frac {b (a-c)+a (b-c) \tan ^2\left (\frac {1}{2} \sqrt {c-a} \sqrt {b-c} \left (\int _1^xf(K[2])dK[2]+c_1\right )\right )}{(b-c) \tan ^2\left (\frac {1}{2} \sqrt {c-a} \sqrt {b-c} \left (\int _1^xf(K[2])dK[2]+c_1\right )\right )+a-c} \\
\end{align*}