40.4.8 problem 19 (i)

Internal problem ID [6648]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 6. Equations of first order and first degree (Linear equations). Supplemetary problems. Page 39
Problem number : 19 (i)
Date solved : Monday, January 27, 2025 at 02:16:49 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} x y^{\prime }+y-x^{3} y^{6}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 254

dsolve(x*diff(y(x),x)+(y(x)-x^3*y(x)^6)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {2^{{1}/{5}} \left (x^{2} \left (2 c_1 \,x^{2}+5\right )^{4}\right )^{{1}/{5}}}{2 c_1 \,x^{3}+5 x} \\ y &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) 2^{{1}/{5}} \left (x^{2} \left (2 c_1 \,x^{2}+5\right )^{4}\right )^{{1}/{5}}}{8 c_1 \,x^{3}+20 x} \\ y &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right ) 2^{{1}/{5}} \left (x^{2} \left (2 c_1 \,x^{2}+5\right )^{4}\right )^{{1}/{5}}}{8 c_1 \,x^{3}+20 x} \\ y &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right ) 2^{{1}/{5}} \left (x^{2} \left (2 c_1 \,x^{2}+5\right )^{4}\right )^{{1}/{5}}}{8 c_1 \,x^{3}+20 x} \\ y &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) 2^{{1}/{5}} \left (x^{2} \left (2 c_1 \,x^{2}+5\right )^{4}\right )^{{1}/{5}}}{8 c_1 \,x^{3}+20 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.089 (sec). Leaf size: 141

DSolve[x*D[y[x],x]+(y[x]-x^3*y[x]^6)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt [5]{-2}}{\sqrt [5]{x^3 \left (5+2 c_1 x^2\right )}} \\ y(x)\to \frac {1}{\sqrt [5]{\frac {5 x^3}{2}+c_1 x^5}} \\ y(x)\to \frac {(-1)^{2/5}}{\sqrt [5]{\frac {5 x^3}{2}+c_1 x^5}} \\ y(x)\to -\frac {(-1)^{3/5}}{\sqrt [5]{\frac {5 x^3}{2}+c_1 x^5}} \\ y(x)\to \frac {(-1)^{4/5}}{\sqrt [5]{\frac {5 x^3}{2}+c_1 x^5}} \\ y(x)\to 0 \\ \end{align*}