7.5.22 problem 22

Internal problem ID [126]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 22
Date solved : Saturday, March 29, 2025 at 04:33:43 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 109

ode:=x^2*diff(y(x),x)+2*x*y(x) = 5*y(x)^4; 
dsolve(ode,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*}

✓ Mathematica. Time used: 0.797 (sec). Leaf size: 96

ode=x^2*D[y[x],x]+2*x*y[x]==5*y[x]^4; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.908 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + 2*x*y(x) - 5*y(x)**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \sqrt [3]{7} \sqrt [3]{\frac {x}{C_{1} x^{7} + 15}}, \ y{\left (x \right )} = \frac {\sqrt [3]{7} \sqrt [3]{\frac {x}{C_{1} x^{7} + 15}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{7} \sqrt [3]{\frac {x}{C_{1} x^{7} + 15}} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]