11.11 problem 22

Internal problem ID [1200]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number: 22.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }-\left (6 x +4\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.015 (sec). Leaf size: 51

Order:=6; 
dsolve(x^2*(1+x)*diff(y(x),x$2)+x*(1+2*x)*diff(y(x),x)-(4+6*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y \relax (x ) = \frac {c_{1} x^{4} \left (1+\mathrm {O}\left (x^{6}\right )\right )+\left (576 x^{4}+\mathrm {O}\left (x^{6}\right )\right ) \ln \relax (x ) c_{2}+\left (-144+192 x -288 x^{2}+576 x^{3}-576 x^{4}-576 x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 48

AsymptoticDSolveValue[x^2*(1+x)*y''[x]+x*(1+2*x)*y'[x]-(4+6*x)*y[x]==0,y[x],{x,0,5}]
 

\[ y(x)\to c_2 x^2+c_1 \left (\frac {3 x^4-12 x^3+6 x^2-4 x+3}{3 x^2}-4 x^2 \log (x)\right ) \]