67.24.16 problem 34.7 (b)

Internal problem ID [16983]
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 : Thursday, October 02, 2025 at 01:41:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 x y^{\prime }-y \,{\mathrm e}^{x}&=0 \end{align*}

Using series method with expansion around

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

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

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

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

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

ode=D[y[x],{x,2}]+3*x*D[y[x],x]-Exp[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.286 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x*Derivative(y(x), x) - y(x)*exp(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} e^{2 x}}{24} - \frac {x^{4} e^{x}}{4} + \frac {x^{2} e^{x}}{2} + 1\right ) + C_{1} x \left (\frac {x^{2} e^{x}}{6} - \frac {x^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]