73.7.34 problem 34

Internal problem ID [15113]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 34
Date solved : Thursday, March 13, 2025 at 05:40:39 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

\begin{align*} y^{\prime }&=4 y-\frac {16 \,{\mathrm e}^{4 x}}{y^{2}} \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 73

ode:=diff(y(x),x) = 4*y(x)-16*exp(4*x)/y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left ({\mathrm e}^{4 x} \left ({\mathrm e}^{8 x} c_{1} +6\right )\right )^{{1}/{3}} \\ y &= -\frac {\left ({\mathrm e}^{4 x} \left ({\mathrm e}^{8 x} c_{1} +6\right )\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left ({\mathrm e}^{4 x} \left ({\mathrm e}^{8 x} c_{1} +6\right )\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 4.187 (sec). Leaf size: 90

ode=D[y[x],x]==4*y[x]-16*Exp[4*x]/y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to e^{4 x/3} \sqrt [3]{6+c_1 e^{8 x}} \\ y(x)\to -\sqrt [3]{-1} e^{4 x/3} \sqrt [3]{6+c_1 e^{8 x}} \\ y(x)\to (-1)^{2/3} e^{4 x/3} \sqrt [3]{6+c_1 e^{8 x}} \\ \end{align*}

✓ Sympy. Time used: 1.439 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*y(x) + Derivative(y(x), x) + 16*exp(4*x)/y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \sqrt [3]{\left (C_{1} e^{8 x} + 6\right ) e^{4 x}}, \ y{\left (x \right )} = \frac {\sqrt [3]{\left (C_{1} e^{8 x} + 6\right ) e^{4 x}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{\left (C_{1} e^{8 x} + 6\right ) e^{4 x}} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]