56.5.18 problem 18

Internal problem ID [8979]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 18
Date solved : Monday, January 27, 2025 at 05:26:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&={\mathrm e}^{a \cos \left (x \right )} \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 85

Order:=8; 
dsolve(diff(y(x),x$2)+y(x)=exp(a*cos(x)),y(x),type='series',x=0);
 

\[ y = \left (-\frac {1}{720} x^{6}+1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{120} x^{5}-\frac {1}{5040} x^{7}\right ) y^{\prime }\left (0\right )+\frac {{\mathrm e}^{a} x^{2}}{2}+\frac {\left (-a -1\right ) {\mathrm e}^{a} x^{4}}{24}+\frac {\left (3 a^{2}+2 a +1\right ) {\mathrm e}^{a} x^{6}}{720}+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 239

AsymptoticDSolveValue[D[y[x],{x,2}]+y[x]==Exp[a*Cos[x]],y[x],{x,0,"8"-1}]
 

\[ y(x)\to \left (-\frac {x^7}{5040}+\frac {x^5}{120}-\frac {x^3}{6}+x\right ) \left (\frac {1}{120} \left (3 a^2+7 a+1\right ) e^a x^5-\frac {\left (15 a^3+60 a^2+31 a+1\right ) e^a x^7}{5040}-\frac {1}{6} (a+1) e^a x^3+e^a x\right )+\left (-\frac {x^6}{720}+\frac {x^4}{24}-\frac {x^2}{2}+1\right ) \left (-\frac {1}{720} \left (15 a^2+15 a+1\right ) e^a x^6+\frac {\left (105 a^3+210 a^2+63 a+1\right ) e^a x^8}{40320}+\frac {1}{24} (3 a+1) e^a x^4-\frac {e^a x^2}{2}\right )+c_2 \left (-\frac {x^7}{5040}+\frac {x^5}{120}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^6}{720}+\frac {x^4}{24}-\frac {x^2}{2}+1\right ) \]