1.30 problem 26 (b)

Internal problem ID [5826]

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: 26 (b).
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _exact, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {y^{\prime \prime }-4 x y^{\prime }-4 y-{\mathrm e}^{x}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.003 (sec). Leaf size: 56

Order:=8; 
dsolve(diff(y(x),x$2)-4*x*diff(y(x),x)-4*y(x)=exp(x),y(x),type='series',x=0);
 

\[ y \relax (x ) = \left (1+2 x^{2}+2 x^{4}+\frac {4}{3} x^{6}\right ) y \relax (0)+\left (x +\frac {4}{3} x^{3}+\frac {16}{15} x^{5}+\frac {64}{105} x^{7}\right ) D\relax (y )\relax (0)+\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {13 x^{4}}{24}+\frac {17 x^{5}}{120}+\frac {29 x^{6}}{80}+\frac {409 x^{7}}{5040}+O\left (x^{8}\right ) \]

Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 94

AsymptoticDSolveValue[y''[x]-4*x*y'[x]-4*y[x]==Exp[x],y[x],{x,0,7}]
 

\[ y(x)\to \frac {409 x^7}{5040}+\frac {29 x^6}{80}+\frac {17 x^5}{120}+\frac {13 x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+c_2 \left (\frac {64 x^7}{105}+\frac {16 x^5}{15}+\frac {4 x^3}{3}+x\right )+c_1 \left (\frac {4 x^6}{3}+2 x^4+2 x^2+1\right ) \]