18.1.4 problem Problem 14.3 (a)

Internal problem ID [3460]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 14, First order ordinary differential equations. 14.4 Exercises, page 490
Problem number : Problem 14.3 (a)
Date solved : Monday, January 27, 2025 at 07:37:32 AM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y \left (2 x^{2} y^{2}+1\right ) y^{\prime }+x \left (y^{4}+1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 119

dsolve(y(x)*(2*x^2*y(x)^2+1)*diff(y(x),x)+x*(y(x)^4+1)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-2-2 \sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\ y \left (x \right ) &= \frac {\sqrt {-2-2 \sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-1+\sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-1+\sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 13.911 (sec). Leaf size: 197

DSolve[y[x]*(2*x^2*y[x]^2+1)*D[y[x],x]+x*(y[x]^4+1)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-\frac {1+\sqrt {-4 x^4+8 c_1 x^2+1}}{x^2}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-\frac {1+\sqrt {-4 x^4+8 c_1 x^2+1}}{x^2}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {-1+\sqrt {-4 x^4+8 c_1 x^2+1}}{x^2}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {-1+\sqrt {-4 x^4+8 c_1 x^2+1}}{x^2}}}{\sqrt {2}} \\ y(x)\to -\sqrt [4]{-1} \\ y(x)\to \sqrt [4]{-1} \\ y(x)\to -(-1)^{3/4} \\ y(x)\to (-1)^{3/4} \\ \end{align*}