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: 0.797 (sec). Leaf size: 821
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 {a^{2} c -2 a b c +4 a b \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}+c_{1} \sqrt {a b -a c -c b +c^{2}}}-2 a c \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}+c_{1} \sqrt {a b -a c -c b +c^{2}}}+b^{2} c -2 b c \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}+c_{1} \sqrt {a b -a c -c b +c^{2}}}+c \,{\mathrm e}^{2 \left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}+2 c_{1} \sqrt {a b -a c -c b +c^{2}}}}{a^{2}-2 a b +2 a \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}+c_{1} \sqrt {a b -a c -c b +c^{2}}}+b^{2}+2 b \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}+c_{1} \sqrt {a b -a c -c b +c^{2}}}-4 \,{\mathrm e}^{\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}+c_{1} \sqrt {a b -a c -c b +c^{2}}} c +{\mathrm e}^{2 \left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}+2 c_{1} \sqrt {a b -a c -c b +c^{2}}}} y \left (x \right ) = \frac {a^{2} c -2 a b c +4 a b \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}-c_{1} \sqrt {a b -a c -c b +c^{2}}}-2 a c \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}-c_{1} \sqrt {a b -a c -c b +c^{2}}}+b^{2} c -2 b c \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}-c_{1} \sqrt {a b -a c -c b +c^{2}}}+c \,{\mathrm e}^{-2 \left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}-2 c_{1} \sqrt {a b -a c -c b +c^{2}}}}{a^{2}-2 a b +2 a \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}-c_{1} \sqrt {a b -a c -c b +c^{2}}}+b^{2}+2 b \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}-c_{1} \sqrt {a b -a c -c b +c^{2}}}-4 \,{\mathrm e}^{-\left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}-c_{1} \sqrt {a b -a c -c b +c^{2}}} c +{\mathrm e}^{-2 \left (\int f \left (x \right )d x \right ) \sqrt {a b -a c -c b +c^{2}}-2 c_{1} \sqrt {a b -a c -c b +c^{2}}}} \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*}