Internal problem ID [1307]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: 16.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} y^{\prime \prime }+x \left (x +5\right ) y^{\prime }-\left (-3 x +2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 47
Order:=6; dsolve(2*x^2*diff(y(x),x$2)+x*(5+x)*diff(y(x),x)-(2-3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-\frac {1}{2} x +\frac {1}{8} x^{2}-\frac {1}{48} x^{3}+\frac {1}{384} x^{4}-\frac {1}{3840} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1+\frac {1}{3} x +\frac {1}{3} x^{2}-\frac {1}{3} x^{3}+\frac {1}{9} x^{4}-\frac {1}{45} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 88
AsymptoticDSolveValue[2*x^2*y''[x]+x*(5+x)*y'[x]-(2-3*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{3840}+\frac {x^4}{384}-\frac {x^3}{48}+\frac {x^2}{8}-\frac {x}{2}+1\right )+\frac {c_2 \left (-\frac {x^5}{45}+\frac {x^4}{9}-\frac {x^3}{3}+\frac {x^2}{3}+\frac {x}{3}+1\right )}{x^2} \]