30.5.26 problem 26

Internal problem ID [7525]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.6, Substitutions and Transformations. Exercises. page 76
Problem number : 26
Date solved : Tuesday, September 30, 2025 at 04:42:00 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

\begin{align*} y^{\prime }+y&=\frac {{\mathrm e}^{x}}{y^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 74

ode:=diff(y(x),x)+y(x) = exp(x)/y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (8 \,{\mathrm e}^{-3 x} c_1 +6 \,{\mathrm e}^{x}\right )^{{1}/{3}}}{2} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (8 \,{\mathrm e}^{-3 x} c_1 +6 \,{\mathrm e}^{x}\right )^{{1}/{3}}}{4} \\ y &= \frac {\left (i \sqrt {3}-1\right ) \left (8 \,{\mathrm e}^{-3 x} c_1 +6 \,{\mathrm e}^{x}\right )^{{1}/{3}}}{4} \\ \end{align*}

✓ Mathematica. Time used: 5.894 (sec). Leaf size: 96

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

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

✓ Sympy. Time used: 0.812 (sec). Leaf size: 90

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

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