33.15 problem 977

Internal problem ID [4210]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 33
Problem number: 977.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\[ \boxed {y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2}=-a} \]

Solution by Maple

Time used: 0.109 (sec). Leaf size: 57

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

\begin{align*} y \left (x \right ) = \sqrt {-x^{2}+a} y \left (x \right ) = -\sqrt {-x^{2}+a} y \left (x \right ) = \sqrt {c_{1}^{2}-2 c_{1} x +a} y \left (x \right ) = -\sqrt {c_{1}^{2}-2 c_{1} x +a} \end{align*}

Solution by Mathematica

Time used: 0.569 (sec). Leaf size: 61

DSolve[y[x]^2 (y'[x])^2+2 x y[x] y'[x]+a-y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {a+c_1 (-2 x+c_1)} y(x)\to \sqrt {a+c_1 (-2 x+c_1)} y(x)\to -\sqrt {a} y(x)\to \sqrt {a} \end{align*}