1.6 problem 3.24 (a)

Internal problem ID [4732]

Book: Advanced Mathemtical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section: Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number: 3.24 (a).
ODE order: 2.
ODE degree: 1.

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

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

Solution by Maple

Time used: 0.021 (sec). Leaf size: 43

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

\[ y \relax (x ) = \left (-\frac {5}{2} x -\frac {3}{8} x^{2}+\frac {1}{12} x^{3}-\frac {5}{192} x^{4}+\frac {3}{320} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}+\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+x +\mathrm {O}\left (x^{6}\right )\right ) \]

Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 53

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

\[ y(x)\to c_2 \left (\frac {3 x^5}{320}-\frac {5 x^4}{192}+\frac {x^3}{12}-\frac {3 x^2}{8}-\frac {5 x}{2}+(x+1) \log (x)\right )+c_1 (x+1) \]