15.48 problem 44

Internal problem ID [1396]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number: 44.
ODE order: 2.
ODE degree: 1.

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

Solve \begin {gather*} \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }+3 y^{\prime } x +\left (4 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.016 (sec). Leaf size: 45

Order:=6; 
dsolve(x^2*(1-2*x)*diff(y(x),x$2)+3*x*diff(y(x),x)+(1+4*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y \relax (x ) = \frac {\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1+\mathrm {O}\left (x^{6}\right )\right )+\left (\left (-6\right ) x +6 x^{2}-\frac {8}{3} x^{3}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x} \]

Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 40

AsymptoticDSolveValue[x^2*(1-2*x)*y''[x]+3*x*y'[x]+(1+4*x)*y[x]==0,y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {-\frac {8 x^3}{3}+6 x^2-6 x}{x}+\frac {\log (x)}{x}\right )+\frac {c_1}{x} \]