Internal problem ID [4166]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 932.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
\[ \boxed {4 x \left (-x +a \right ) \left (-x +b \right ) {y^{\prime }}^{2}=\left (a b -2 x \left (a +b \right )+2 x^{2}\right )^{2}} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 81
dsolve(4*x*(a-x)*(b-x)*diff(y(x),x)^2 = (a*b-2*x*(a+b)+2*x^2)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \int -\frac {a b -2 a x -2 x b +2 x^{2}}{2 \sqrt {x \left (-x +b \right ) \left (-x +a \right )}}d x +c_{1} y \left (x \right ) = \int \frac {a b -2 a x -2 x b +2 x^{2}}{2 \sqrt {x \left (-x +b \right ) \left (-x +a \right )}}d x +c_{1} \end{align*}
✓ Solution by Mathematica
Time used: 14.208 (sec). Leaf size: 375
DSolve[4 x(a-x)(b-x) (y'[x])^2==(a b-2 x(a+b)+2 x^2)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1-\frac {(a-x) \left (2 \left (a^2-b^2\right ) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} E\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+b (a+2 b) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right ),\frac {a}{a-b}\right )+2 i x \sqrt {\frac {x}{a}-1} (b-x)\right )}{3 \sqrt {\frac {x}{a}-1} \sqrt {x (a-x) (x-b)}} y(x)\to \frac {(a-x) \left (2 \left (a^2-b^2\right ) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} E\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+b (a+2 b) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right ),\frac {a}{a-b}\right )+2 i x \sqrt {\frac {x}{a}-1} (b-x)\right )}{3 \sqrt {\frac {x}{a}-1} \sqrt {x (a-x) (x-b)}}+c_1 \end{align*}