52.1.26 problem 24

Internal problem ID [8243]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 24
Date solved : Sunday, March 30, 2025 at 12:49:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+{\mathrm e}^{x} y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 74

Order:=8; 
ode:=diff(diff(y(x),x),x)+exp(x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{240} x^{6}+\frac {1}{840} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}-\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 91

ode=D[y[x],{x,2}]+Exp[x]*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^7}{840}+\frac {x^6}{240}-\frac {x^5}{120}-\frac {x^3}{6}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^7}{5040}-\frac {x^6}{720}+\frac {x^5}{120}-\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 1.781 (sec). Leaf size: 182

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + exp(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6} e^{4 x}}{720} + \frac {x^{6} e^{2 x}}{240} + \frac {x^{6}}{720} - \frac {x^{5} e^{3 x}}{120} - \frac {x^{5} e^{x}}{60} + \frac {x^{4} e^{2 x}}{24} + \frac {x^{4}}{24} - \frac {x^{3} e^{x}}{6} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (- \frac {x^{5} e^{5 x}}{720} - \frac {x^{5} e^{3 x}}{180} - \frac {x^{5} e^{x}}{240} + \frac {x^{4} e^{4 x}}{120} + \frac {x^{4} e^{2 x}}{40} + \frac {x^{4}}{120} - \frac {x^{3} e^{3 x}}{24} - \frac {x^{3} e^{x}}{12} + \frac {x^{2} e^{2 x}}{6} + \frac {x^{2}}{6} - \frac {x e^{x}}{2} + 1\right ) + O\left (x^{8}\right ) \]