Internal problem ID [6481]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (\cos \relax (x )-1\right ) y^{\prime }+{\mathrm e}^{x} y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.138 (sec). Leaf size: 1171
Order:=6; dsolve(x^2*diff(y(x), x$2) + (cos(x)-1)*diff(y(x), x) + exp(x)*y(x) = 0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{4} i \sqrt {3} x +\frac {-i \sqrt {3}+11}{32 i \sqrt {3}-64} x^{2}-\frac {55}{288} \frac {i \sqrt {3}+3}{\left (i \sqrt {3}-3\right ) \left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{3}-\frac {1}{384} \frac {112 i \sqrt {3}-199}{\left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-3\right ) \left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{4}+\frac {\frac {18491 i \sqrt {3}}{38400}+\frac {4387}{12800}}{\left (i \sqrt {3}-5\right ) \left (i \sqrt {3}-4\right ) \left (i \sqrt {3}-3\right ) \left (-2+i \sqrt {3}\right ) \left (-1+i \sqrt {3}\right )} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{4} i \sqrt {3} x +\frac {-i \sqrt {3}-11}{32 i \sqrt {3}+64} x^{2}-\frac {55}{288} \frac {i \sqrt {3}-3}{\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{3}+\frac {1}{384} \frac {112 i \sqrt {3}+199}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{4}+\frac {\frac {18491 i \sqrt {3}}{38400}-\frac {4387}{12800}}{\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (1+i \sqrt {3}\right )} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 2502
AsymptoticDSolveValue[x^2*y''[x] + (Cos[x]-1)*y'[x] + Exp[x]*y[x] ==0,y[x],{x,0,5}]
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