Internal problem ID [5901]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left ({\mathrm e}^{x}-1-x \right ) y^{\prime \prime }+x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.154 (sec). Leaf size: 70
Order:=8; dsolve((exp(x)-1-x)*diff(y(x),x$2)+x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-x +\frac {4}{9} x^{2}-\frac {29}{216} x^{3}+\frac {37}{1200} x^{4}-\frac {58}{10125} x^{5}+\frac {14209}{15876000} x^{6}-\frac {107329}{889056000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (\left (-2\right ) x +2 x^{2}-\frac {8}{9} x^{3}+\frac {29}{108} x^{4}-\frac {37}{600} x^{5}+\frac {116}{10125} x^{6}-\frac {14209}{7938000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (1-\frac {8}{3} x^{2}+\frac {175}{108} x^{3}-\frac {3727}{6480} x^{4}+\frac {47531}{324000} x^{5}-\frac {3003737}{102060000} x^{6}+\frac {48833381}{10001880000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.349 (sec). Leaf size: 133
AsymptoticDSolveValue[(Exp[x]-1-x)*y''[x]+x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (x^6 \left (\frac {116 \log (x)}{10125}-\frac {3003737}{102060000}\right )+x^5 \left (\frac {47531}{324000}-\frac {37 \log (x)}{600}\right )+x^4 \left (\frac {29 \log (x)}{108}-\frac {3727}{6480}\right )+x^3 \left (\frac {175}{108}-\frac {8 \log (x)}{9}\right )+x^2 \left (2 \log (x)-\frac {8}{3}\right )-2 x \log (x)+1\right )+c_2 x \left (-\frac {107329 x^7}{889056000}+\frac {14209 x^6}{15876000}-\frac {58 x^5}{10125}+\frac {37 x^4}{1200}-\frac {29 x^3}{216}+\frac {4 x^2}{9}-x+1\right ) \]