3.5 problem 5

Internal problem ID [4192]

Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section: Chapter VII, Solutions in series. Examples XIV. page 177
Problem number: 5.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]

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

Solution by Maple

Time used: 0.0 (sec). Leaf size: 30

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

\[ y \relax (x ) = \left (1-\frac {a \,x^{4}}{12}\right ) y \relax (0)+\left (x -\frac {1}{20} a \,x^{5}\right ) D\relax (y )\relax (0)+\frac {x^{2}}{2}+\frac {x^{3}}{6}+O\left (x^{6}\right ) \]

Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 44

AsymptoticDSolveValue[y''[x]+a*x^2*y[x]==1+x,y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (x-\frac {a x^5}{20}\right )+c_1 \left (1-\frac {a x^4}{12}\right )+\frac {x^3}{6}+\frac {x^2}{2} \]