Internal problem ID [7234]
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]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (\cos \left (x \right )-1\right ) y^{\prime }+{\mathrm e}^{x} y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.125 (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);
\[ y \left (x \right ) = \sqrt {x}\, \left (c_{2} x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{4} i \sqrt {3} x +\frac {-i \sqrt {3}-11}{32 i \sqrt {3}+64} x^{2}+\frac {\frac {55 \sqrt {3}}{288}+\frac {55 i}{96}}{\left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right )} x^{3}+\frac {1}{384} \frac {112 i \sqrt {3}+199}{\left (-\sqrt {3}+2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right )} x^{4}+\frac {\frac {18491 \sqrt {3}}{38400}+\frac {4387 i}{12800}}{\left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+5\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{4} i \sqrt {3} x +\frac {-\sqrt {3}-11 i}{32 \sqrt {3}+64 i} x^{2}+\frac {55 \sqrt {3}-165 i}{3456 i-2304 \sqrt {3}} x^{3}+\frac {199 i+112 \sqrt {3}}{-27648 i+7680 \sqrt {3}} x^{4}+\frac {\frac {18491 \sqrt {3}}{38400}-\frac {4387 i}{12800}}{\left (\sqrt {3}+i\right ) \left (\sqrt {3}+2 i\right ) \left (\sqrt {3}+3 i\right ) \left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+5 i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]
✓ 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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