83.45.2 problem Ex 2 page 52

Internal problem ID [19545]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter IV. Equations of the first order but not of the first degree
Problem number : Ex 2 page 52
Date solved : Tuesday, January 28, 2025 at 01:50:22 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} x^{2} \left ({y^{\prime }}^{2}-y^{2}\right )+y^{2}&=x^{4}+2 x y y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 3.346 (sec). Leaf size: 56

dsolve(x^2*(diff(y(x),x)^2-y(x)^2)+y(x)^2=x^4+2*x*y(x)*diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= -\frac {x \left ({\mathrm e}^{x}-c_{1}^{2} {\mathrm e}^{-x}\right )}{2 c_{1}} \\ y \left (x \right ) &= \frac {x \left ({\mathrm e}^{x} c_{1}^{2}-{\mathrm e}^{-x}\right )}{2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.143 (sec). Leaf size: 26

DSolve[x^2*(D[y[x],x]^2-y[x]^2)+y[x]^2==x^4+2*x*y[x]*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -x \sinh (x-c_1) \\ y(x)\to x \sinh (x+c_1) \\ \end{align*}