67.24.6 problem 34.5 (f)

Internal problem ID [16973]
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 : 34.5 (f)
Date solved : Thursday, October 02, 2025 at 01:41:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} {\mathrm e}^{3 x} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\frac {2 y}{x^{2}+4}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 54

Order:=6; 
ode:=exp(3*x)*diff(diff(y(x),x),x)+sin(x)*diff(y(x),x)+2/(x^2+4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {1}{8} x^{4}-\frac {7}{160} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {3}{8} x^{4}-\frac {67}{240} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 63

ode=Exp[3*x]*D[y[x],{x,2}]+Sin[x]*D[y[x],x]+2/(x^2+4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {67 x^5}{240}+\frac {3 x^4}{8}-\frac {x^3}{4}+x\right )+c_1 \left (-\frac {7 x^5}{160}-\frac {x^4}{8}+\frac {x^3}{4}-\frac {x^2}{4}+1\right ) \]

✓ Sympy. Time used: 2.660 (sec). Leaf size: 139

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(exp(3*x)*Derivative(y(x), (x, 2)) + sin(x)*Derivative(y(x), x) + 2*y(x)/(x**2 + 4),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} e^{- 3 x}}{96} + \frac {x^{4} e^{- 6 x}}{96} - \frac {x^{4} e^{- 9 x} \sin ^{2}{\left (x \right )}}{48} + \frac {x^{3} e^{- 6 x} \sin {\left (x \right )}}{12} - \frac {x^{2} e^{- 3 x}}{4} + 1\right ) + C_{1} x \left (\frac {x^{3} e^{- 6 x} \sin {\left (x \right )}}{24} - \frac {x^{3} e^{- 9 x} \sin ^{3}{\left (x \right )}}{24} - \frac {x^{2} e^{- 3 x}}{12} + \frac {x^{2} e^{- 6 x} \sin ^{2}{\left (x \right )}}{6} - \frac {x e^{- 3 x} \sin {\left (x \right )}}{2} + 1\right ) + O\left (x^{6}\right ) \]