85.60.3 problem 5

Internal problem ID [22854]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. B Exercises at page 203
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:15:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 25

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

\[ y = x \left (\operatorname {Ei}_{1}\left (x \right ) x +c_2 x +\operatorname {Ei}_{1}\left (x \right )-{\mathrm e}^{-x}+c_1 \right ) \]

✓ Mathematica. Time used: 0.756 (sec). Leaf size: 411

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

\begin{align*} y(x)&\to 2^{-\frac {1}{2} i \left (\sqrt {7}-i\right )} x^{\frac {1}{2}-\frac {i \sqrt {7}}{2}} \left (x^{i \sqrt {7}} \operatorname {Hypergeometric1F1}\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2},1-i \sqrt {7},-\frac {2}{x}\right ) \int _1^x-i \sqrt {\frac {2}{7}} e^{-K[2]+\frac {1}{2} i \sqrt {7} \log (2)+\frac {2}{K[2]}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-\frac {2}{K[2]}\right ) K[2]^{-\frac {1}{2}-\frac {i \sqrt {7}}{2}}dK[2]+2^{i \sqrt {7}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-\frac {2}{x}\right ) \int _1^xi \sqrt {\frac {2}{7}} e^{-K[1]-\frac {1}{2} i \sqrt {7} \log (2)+\frac {2}{K[1]}} \operatorname {Hypergeometric1F1}\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2},1-i \sqrt {7},-\frac {2}{K[1]}\right ) K[1]^{\frac {1}{2} i \left (i+\sqrt {7}\right )}dK[1]+c_2 x^{i \sqrt {7}} \operatorname {Hypergeometric1F1}\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2},1-i \sqrt {7},-\frac {2}{x}\right )+2^{i \sqrt {7}} c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-\frac {2}{x}\right )\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x**2*Derivative(y(x), (x, 2))/2 + x*exp(-x)/2 - y(x) + Derivati