29.33.18 problem 980

Internal problem ID [5557]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 980
Date solved : Monday, January 27, 2025 at 11:48:12 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (a -1\right ) b +a \,x^{2}+\left (1-a \right ) y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.134 (sec). Leaf size: 88

dsolve(y(x)^2*diff(y(x),x)^2+2*a*x*y(x)*diff(y(x),x)+(a-1)*b+a*x^2+(1-a)*y(x)^2 = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.197 (sec). Leaf size: 65

DSolve[y[x]^2 (D[y[x],x])^2+2 a x y[x] D[y[x],x]+(a-1)b+a x^2+(1-a)y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-2 (a-1) c_1 x+(a-1) c_1{}^2+b-x^2} \\ y(x)\to \sqrt {-2 (a-1) c_1 x+(a-1) c_1{}^2+b-x^2} \\ \end{align*}