52.4.12 problem 20

Internal problem ID [8322]
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. CHAPTER 6 IN REVIEW. Page 271
Problem number : 20
Date solved : Wednesday, March 05, 2025 at 05:35:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left ({\mathrm e}^{x}-1-x \right ) y^{\prime \prime }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.166 (sec). Leaf size: 70

Order:=8; 
ode:=(exp(x)-1-x)*diff(diff(y(x),x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x \left (1-x +\frac {4}{9} x^{2}-\frac {29}{216} x^{3}+\frac {37}{1200} x^{4}-\frac {58}{10125} x^{5}+\frac {14209}{15876000} x^{6}-\frac {107329}{889056000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-2\right ) x +2 x^{2}-\frac {8}{9} x^{3}+\frac {29}{108} x^{4}-\frac {37}{600} x^{5}+\frac {116}{10125} x^{6}-\frac {14209}{7938000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {8}{3} x^{2}+\frac {175}{108} x^{3}-\frac {3727}{6480} x^{4}+\frac {47531}{324000} x^{5}-\frac {3003737}{102060000} x^{6}+\frac {48833381}{10001880000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.391 (sec). Leaf size: 133

ode=(Exp[x]-1-x)*D[y[x],{x,2}]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (x^6 \left (\frac {116 \log (x)}{10125}-\frac {3003737}{102060000}\right )+x^5 \left (\frac {47531}{324000}-\frac {37 \log (x)}{600}\right )+x^4 \left (\frac {29 \log (x)}{108}-\frac {3727}{6480}\right )+x^3 \left (\frac {175}{108}-\frac {8 \log (x)}{9}\right )+x^2 \left (2 \log (x)-\frac {8}{3}\right )-2 x \log (x)+1\right )+c_2 x \left (-\frac {107329 x^7}{889056000}+\frac {14209 x^6}{15876000}-\frac {58 x^5}{10125}+\frac {37 x^4}{1200}-\frac {29 x^3}{216}+\frac {4 x^2}{9}-x+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + (-x + exp(x) - 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : ODE x*y(x) + (-x + exp(x) - 1)*Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular