60.1.488 problem 491

Internal problem ID [10502]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 491
Date solved : Monday, January 27, 2025 at 08:24:11 PM
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 (1-a \right ) y^{2}+a \,x^{2}+\left (a -1\right ) b&=0 \end{align*}

✓ Solution by Maple

Time used: 0.136 (sec). Leaf size: 88

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

\begin{align*} y &= \sqrt {-a \,x^{2}+b} \\ y &= -\sqrt {-a \,x^{2}+b} \\ y &= \sqrt {a \,c_{1}^{2}-2 a c_{1} x -c_{1}^{2}+2 c_{1} x -x^{2}+b} \\ y &= -\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: 1.224 (sec). Leaf size: 65

DSolve[(-1 + a)*b + a*x^2 + (1 - a)*y[x]^2 + 2*a*x*y[x]*D[y[x],x] + y[x]^2*D[y[x],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*}