5.3 problem 8

Internal problem ID [4219]

Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section: Chapter VII, Solutions in series. Examples XVI. page 220
Problem number: 8.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [_Jacobi]

Solve \begin {gather*} \boxed {2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.062 (sec). Leaf size: 44

Order:=6; 
dsolve(2*x*(1-x)*diff(y(x),x$2)+(1-11*x)*diff(y(x),x)-10*y(x)=0,y(x),type='series',x=0);
 

\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+5 x +14 x^{2}+30 x^{3}+55 x^{4}+91 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1+10 x +35 x^{2}+84 x^{3}+165 x^{4}+286 x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]

Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 65

AsymptoticDSolveValue[2*x*(1-x)*y''[x]+(1-11*x)*y'[x]-10*y[x]==0,y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (91 x^5+55 x^4+30 x^3+14 x^2+5 x+1\right )+c_2 \left (286 x^5+165 x^4+84 x^3+35 x^2+10 x+1\right ) \]