Internal problem ID [6558]
Book: Own collection of miscellaneous problems
Section: section 5.0
Problem number: 18.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y-{\mathrm e}^{a \cos \relax (x )}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 76
Order:=8; dsolve(diff(y(x),x$2)+y(x)=exp(a*cos(x)),y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}\right ) y \relax (0)+\left (x -\frac {1}{6} x^{3}+\frac {1}{120} x^{5}-\frac {1}{5040} x^{7}\right ) D\relax (y )\relax (0)+\frac {{\mathrm e}^{a} x^{2}}{2}+\frac {\left (-1-a \right ) {\mathrm e}^{a} x^{4}}{24}+\frac {\left (3 a^{2}+2 a +1\right ) {\mathrm e}^{a} x^{6}}{720}+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.06 (sec). Leaf size: 239
AsymptoticDSolveValue[y''[x]+y[x]==Exp[a*Cos[x]],y[x],{x,0,7}]
\[ y(x)\to \left (-\frac {x^7}{5040}+\frac {x^5}{120}-\frac {x^3}{6}+x\right ) \left (\frac {1}{120} \left (3 a^2+7 a+1\right ) e^a x^5-\frac {\left (15 a^3+60 a^2+31 a+1\right ) e^a x^7}{5040}-\frac {1}{6} (a+1) e^a x^3+e^a x\right )+\left (-\frac {x^6}{720}+\frac {x^4}{24}-\frac {x^2}{2}+1\right ) \left (-\frac {1}{720} \left (15 a^2+15 a+1\right ) e^a x^6+\frac {\left (105 a^3+210 a^2+63 a+1\right ) e^a x^8}{40320}+\frac {1}{24} (3 a+1) e^a x^4-\frac {e^a x^2}{2}\right )+c_2 \left (-\frac {x^7}{5040}+\frac {x^5}{120}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^6}{720}+\frac {x^4}{24}-\frac {x^2}{2}+1\right ) \]