27.4 problem 770

Internal problem ID [4010]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 27
Problem number: 770.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [_separable]

\[ \boxed {{y^{\prime }}^{2}-f \left (x \right )^{2} \left (y-a \right ) \left (y-b \right ) \left (y-c \right )^{2}=0} \]

Solution by Maple

Time used: 1.157 (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 (a -c \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 (a -c \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 (a -c \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 (a -c \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 (a -c \right ) \left (b -c \right )}}+\left ({\mathrm e}^{-2 \left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \right ) \left (b -c \right )}}+\left (a -b \right )^{2}\right ) c}{\left (2 a +2 b -4 c \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x +c_{1} \right ) \sqrt {\left (a -c \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 (a -c \right ) \left (b -c \right )}}} \\ \end{align*}

Solution by Mathematica

Time used: 60.31 (sec). Leaf size: 223

DSolve[(y'[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*}