29.26.26 problem 762

Internal problem ID [5347]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 762
Date solved : Monday, January 27, 2025 at 11:16:07 AM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}&=\left (y-a \right ) \left (y-b \right ) \left (y-c \right ) \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 75

dsolve(diff(y(x),x)^2 = (y(x)-a)*(y(x)-b)*(y(x)-c),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= a \\ y \left (x \right ) &= b \\ y \left (x \right ) &= c \\ x -\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} -c_{1} &= 0 \\ x +\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} -c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 33.309 (sec). Leaf size: 188

DSolve[(D[y[x],x])^2==(y[x]-a)(y[x]-b)*(y[x]-c),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} (c_1-i x)|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} (c_1-i x)|\frac {a-c}{a-b}\right ){}^2-a+b\right ) \\ y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} (i x+c_1)|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} (i x+c_1)|\frac {a-c}{a-b}\right ){}^2-a+b\right ) \\ y(x)\to a \\ y(x)\to b \\ y(x)\to c \\ \end{align*}