20.24.20 problem Problem 21

Internal problem ID [4005]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 21
Date solved : Monday, January 27, 2025 at 08:06:03 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+x y^{\prime }-4 y&=6 \,{\mathrm e}^{x} \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 57

Order:=6; 
dsolve(diff(y(x),x$2)+x*diff(y(x),x)-4*y(x)=6*exp(x),y(x),type='series',x=0);
 

\[ y \left (x \right ) = \left (1+2 x^{2}+\frac {1}{3} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}+\frac {1}{40} x^{5}\right ) D\left (y \right )\left (0\right )+3 x^{2}+x^{3}+\frac {3 x^{4}}{4}+\frac {x^{5}}{10}+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 62

AsymptoticDSolveValue[D[y[x],{x,2}]+x*D[y[x],x]-4*y[x]==6*Exp[x],y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {x^5}{10}+\frac {3 x^4}{4}+x^3+3 x^2+c_2 \left (\frac {x^5}{40}+\frac {x^3}{2}+x\right )+c_1 \left (\frac {x^4}{3}+2 x^2+1\right ) \]