43.2.9 problem 7.3.8 (d)

Internal problem ID [6866]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page 300
Problem number : 7.3.8 (d)
Date solved : Monday, January 27, 2025 at 02:31:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 54

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

\[ y = c_1 x \left (1+\frac {1}{6} x^{2}+\frac {1}{72} x^{3}+\frac {7}{480} x^{4}+\frac {29}{10800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x +\frac {1}{6} x^{3}+\frac {1}{72} x^{4}+\frac {7}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-x -\frac {2}{9} x^{3}-\frac {11}{864} x^{4}-\frac {109}{4800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 70

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

\[ y(x)\to c_2 \left (\frac {7 x^5}{480}+\frac {x^4}{72}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {1}{864} \left (-23 x^4-336 x^3-1728 x+864\right )+\frac {1}{72} x \left (x^3+12 x^2+72\right ) \log (x)\right ) \]