Internal problem ID [13630]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {{\mathrm e}^{3 x} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\frac {2 y}{x^{2}+4}=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
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);
\[ y \left (x \right ) = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {1}{8} x^{4}-\frac {7}{160} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {3}{8} x^{4}-\frac {67}{240} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 63
AsymptoticDSolveValue[Exp[3*x]*y''[x]+Sin[x]*y'[x]+2/(x^2+4)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (-\frac {67 x^5}{240}+\frac {3 x^4}{8}-\frac {x^3}{4}+x\right )+c_1 \left (-\frac {7 x^5}{160}-\frac {x^4}{8}+\frac {x^3}{4}-\frac {x^2}{4}+1\right ) \]