29.25.16 problem 713

Internal problem ID [5303]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 713
Date solved : Monday, January 27, 2025 at 11:07:50 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 2 x \left (x^{3}+y^{4}\right ) y^{\prime }&=\left (x^{3}+2 y^{4}\right ) y \end{align*}

✓ Solution by Maple

Time used: 0.316 (sec). Leaf size: 281

dsolve(2*x*(x^3+y(x)^4)*diff(y(x),x) = (x^3+2*y(x)^4)*y(x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {2^{{3}/{4}} \left (\left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {2^{{3}/{4}} \left (\left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {2^{{3}/{4}} \left (\left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {2^{{3}/{4}} \left (\left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {i 2^{{3}/{4}} \left (\left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {i 2^{{3}/{4}} \left (\left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {i 2^{{3}/{4}} \left (\left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {i 2^{{3}/{4}} \left (\left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.898 (sec). Leaf size: 166

DSolve[2 x(x^3+y[x]^4)D[y[x],x]==(x^3+2 y[x]^4)y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {c_1 x^2-x^{3/2} \sqrt {4+c_1{}^2 x}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {c_1 x^2-x^{3/2} \sqrt {4+c_1{}^2 x}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {x^{3/2} \sqrt {4+c_1{}^2 x}+c_1 x^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {x^{3/2} \sqrt {4+c_1{}^2 x}+c_1 x^2}}{\sqrt {2}} \\ y(x)\to 0 \\ \end{align*}