83.45.18 problem Ex 18 page 60

Internal problem ID [19561]
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 18 page 60
Date solved : Tuesday, January 28, 2025 at 01:53:53 PM
CAS classification : [_separable]

\begin{align*} x y^{2} \left ({y^{\prime }}^{2}+2\right )&=2 y^{\prime } y^{3}+x^{3} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 52

dsolve(x*y(x)^2*(diff(y(x),x)^2+2)=2*diff(y(x),x)*y(x)^3+x^3,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.633 (sec). Leaf size: 85

DSolve[x*y[x]^2*(D[y[x],x]^2+2)==2*D[y[x],x]*y[x]^3+x^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {x^2+2 c_1} \\ y(x)\to \sqrt {x^2+2 c_1} \\ y(x)\to -\sqrt {x^2+c_1 x^4} \\ y(x)\to \sqrt {x^2+c_1 x^4} \\ y(x)\to -x \\ y(x)\to x \\ \end{align*}