Internal problem ID [1332]
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: 43.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 32
Order:=6; dsolve(x*(1+x^2)*diff(y(x),x$2)+(4+7*x^2)*diff(y(x),x)+8*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \left (1-\frac {4}{5} x^{2}+\frac {24}{35} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-6 x^{2}+\frac {9}{2} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 42
AsymptoticDSolveValue[x*(1+x^2)*y''[x]+(4+7*x^2)*y'[x]+8*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {3 x}{8}-\frac {1}{2 x}\right )+c_2 \left (\frac {24 x^4}{35}-\frac {4 x^2}{5}+1\right ) \]