73.24.6 problem 34.5 (f)

Internal problem ID [15686]
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 : Tuesday, January 28, 2025 at 08:05:13 AM
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*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 54

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 63

AsymptoticDSolveValue[Exp[3*x]*D[y[x],{x,2}]+Sin[x]*D[y[x],x]+2/(x^2+4)*y[x]==0,y[x],{x,0,"6"-1}]
 

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