29.27.1 problem 766

Internal problem ID [5351]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 27
Problem number : 766
Date solved : Monday, January 27, 2025 at 11:16:53 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 )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.143 (sec). Leaf size: 112

dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)^2*(y(x)-b) = 0,y(x), singsol=all)
 

\begin{align*} \frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}+\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}-\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.102 (sec). Leaf size: 93

DSolve[(D[y[x],x])^2+f[x]*(y[x]-a)^2 *(y[x]-b)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to b+(b-a) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \left (\int _1^x-\sqrt {f(K[1])}dK[1]+c_1\right )\right ) \\ y(x)\to b+(b-a) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \left (\int _1^x\sqrt {f(K[2])}dK[2]+c_1\right )\right ) \\ \end{align*}