Internal problem ID [4191]
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: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+2 y^{\prime }+a^{3} x^{2} y-2=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 50
Order:=6; dsolve(x*diff(y(x),x$2)+2*diff(y(x),x)+a^3*x^2*y(x)=2,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \left (1-\frac {1}{12} a^{3} x^{3}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {1}{6} a^{3} x^{3}+\mathrm {O}\left (x^{6}\right )\right )}{x}+x \left (1-\frac {1}{20} a^{3} x^{3}+\mathrm {O}\left (x^{5}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 136
AsymptoticDSolveValue[x*y''[x]+2*y'[x]+a^3*x^2*y[x]==2,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {a^6 x^6}{504}-\frac {a^3 x^3}{12}+1\right )+\frac {c_2 \left (\frac {a^6 x^6}{180}-\frac {a^3 x^3}{6}+1\right )}{x}+\left (2 x-\frac {a^3 x^4}{12}\right ) \left (\frac {a^6 x^6}{504}-\frac {a^3 x^3}{12}+1\right )+\frac {\left (\frac {a^3 x^5}{30}-x^2\right ) \left (\frac {a^6 x^6}{180}-\frac {a^3 x^3}{6}+1\right )}{x} \]