68.18.58 problem 64

Internal problem ID [17902]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 64
Date solved : Thursday, October 02, 2025 at 02:29:27 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 69

Order:=6; 
ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)-3*y(x) = x*exp(x); 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {3}{2} x^{2}-x^{3}+\frac {7}{8} x^{4}-\frac {1}{2} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {7}{6} x^{3}-\frac {5}{6} x^{4}+\frac {61}{120} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}+\frac {x^{5}}{20}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 66

ode=D[y[x],{x,2}]+2*D[y[x],x]-3*y[x]==x*Exp[x]; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{2}+\frac {7 x^4}{8}-x^3+\frac {3 x^2}{2}+1\right )+c_2 \left (\frac {61 x^5}{120}-\frac {5 x^4}{6}+\frac {7 x^3}{6}-x^2+x\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(x) - 3*y(x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE -x*exp(x) - 3*y(x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular