8.5.22 problem 22

Internal problem ID [750]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 22
Date solved : Wednesday, February 05, 2025 at 03:58:03 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} 2 x y+x^{2} y^{\prime }&=5 y^{4} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 109

dsolve(2*x*y(x)+x^2*diff(y(x),x) = 5*y(x)^4,y(x), singsol=all)
 

\begin{align*} y &= \frac {7^{{1}/{3}} {\left (x \left (7 c_1 \,x^{7}+15\right )^{2}\right )}^{{1}/{3}}}{7 c_1 \,x^{7}+15} \\ y &= -\frac {7^{{1}/{3}} {\left (x \left (7 c_1 \,x^{7}+15\right )^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{14 c_1 \,x^{7}+30} \\ y &= \frac {7^{{1}/{3}} {\left (x \left (7 c_1 \,x^{7}+15\right )^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{14 c_1 \,x^{7}+30} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.714 (sec). Leaf size: 96

DSolve[2*x*y[x]+x^2*D[y[x],x] == 5*y[x]^4,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt [3]{-7} \sqrt [3]{x}}{\sqrt [3]{15+7 c_1 x^7}} \\ y(x)\to \frac {\sqrt [3]{7} \sqrt [3]{x}}{\sqrt [3]{15+7 c_1 x^7}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{7} \sqrt [3]{x}}{\sqrt [3]{15+7 c_1 x^7}} \\ y(x)\to 0 \\ \end{align*}