Internal problem ID [2415]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page
758
Problem number: 9.
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 }-x \cos \relax (x ) y^{\prime }+5 y \,{\mathrm e}^{2 x}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 71
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*cos(x)*diff(y(x),x)+5*exp(2*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{1-2 i} \left (1+\left (-\frac {10}{17}-\frac {40 i}{17}\right ) x +\left (-\frac {365}{136}+\frac {13 i}{17}\right ) x^{2}+\left (\frac {223}{1020}+\frac {1723 i}{765}\right ) x^{3}+\left (\frac {114911}{78336}+\frac {24835 i}{78336}\right ) x^{4}+\left (\frac {4041077}{8029440}-\frac {1112267 i}{1605888}\right ) x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{1+2 i} \left (1+\left (-\frac {10}{17}+\frac {40 i}{17}\right ) x +\left (-\frac {365}{136}-\frac {13 i}{17}\right ) x^{2}+\left (\frac {223}{1020}-\frac {1723 i}{765}\right ) x^{3}+\left (\frac {114911}{78336}-\frac {24835 i}{78336}\right ) x^{4}+\left (\frac {4041077}{8029440}+\frac {1112267 i}{1605888}\right ) x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 94
AsymptoticDSolveValue[x^2*y''[x]-x*Cos[x]*y'[x]+5*Exp[2*x]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {11}{391680}+\frac {7 i}{391680}\right ) c_1 \left ((32064-31693 i) x^4-(30784+60608 i) x^3-(80352-23904 i) x^2+(23040+69120 i) x+(25344-16128 i)\right ) x^{1+2 i}+\left (\frac {7}{391680}+\frac {11 i}{391680}\right ) c_2 \left ((31693-32064 i) x^4+(60608+30784 i) x^3-(23904-80352 i) x^2-(69120+23040 i) x+(16128-25344 i)\right ) x^{1-2 i} \]