38.6.46 problem 46

Internal problem ID [8476]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.3 Linear equations. Exercises 2.3 at page 63
Problem number : 46
Date solved : Tuesday, September 30, 2025 at 05:37:48 PM
CAS classification : [_linear]

\begin{align*} x^{2} y^{\prime }-y&=x^{3} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 46

ode:=x^2*diff(y(x),x)-y(x) = x^3; 
ic:=[y(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{2}}{2}+\frac {x}{2}+\frac {{\mathrm e}^{-\frac {1}{x}} \operatorname {Ei}_{1}\left (-\frac {1}{x}\right )}{2}+\frac {{\mathrm e}^{-\frac {1}{x}} \left (-\operatorname {Ei}_{1}\left (-1\right )-2 \,{\mathrm e}\right )}{2} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 29

ode=x^2*D[y[x],x]-y[x]==x^3; 
ic={y[1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-1/x} \int _1^xe^{\frac {1}{K[1]}} K[1]dK[1] \end{align*}

✓ Sympy. Time used: 0.694 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3 + x**2*Derivative(y(x), x) - y(x),0) 
ics = {y(1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (x^{2} \operatorname {E}_{3}\left (- \frac {1}{x}\right ) - \operatorname {E}_{3}\left (-1\right )\right ) e^{- \frac {1}{x}} \]