73.24.16 problem 34.7 (b)

Internal problem ID [15696]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.7 (b)
Date solved : Tuesday, January 28, 2025 at 08:05:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 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: 0.004 (sec). Leaf size: 54

Order:=6; 
dsolve(diff(y(x),x$2)+3*x*diff(y(x),x)-exp(x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{6} x^{4}-\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {19}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 63

AsymptoticDSolveValue[D[y[x],{x,2}]+3*x*D[y[x],x]-Exp[x]*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {19 x^5}{120}+\frac {x^4}{12}-\frac {x^3}{3}+x\right )+c_1 \left (-\frac {x^5}{30}-\frac {x^4}{6}+\frac {x^3}{6}+\frac {x^2}{2}+1\right ) \]