2.1033   ODE No. 1033

  1. Problem in Latex
  2. Mathematica input
  3. Maple input

ae2xy(x)+y(x)+y(x)=0 Mathematica : cpu = 0.0181377 (sec), leaf count = 37

{{y(x)c1cos(aex)c2sin(aex)}}

Maple : cpu = 0.01 (sec), leaf count = 27

{y(x)=_C1sin(exa)+_C2cos(exa)}

Hand solution

y+y+ae2xy=0

Let y(x)=η(ξ) where ξ=ex, hence

dydx=dηdξdξdx=dηdξ(ex)

And

d2ydx2=ddx(dηdξ(ex))=d2ηdξ2dξdx(ex)+dηdξ(ex)=d2ηdξ2(ex)(ex)+dηdξ(ex)=d2ηdξ2(e2x)+dηdξ(ex)

Hence the original ODE becomes

d2ηdξ2(e2x)+dηdξ(ex)+dηdξ(ex)+ae2xη(ξ)=0η+aη=0

This is standard second order with constant coefficients. The solution is

η=c1cos(aξ)+c2sin(aξ)

Substituting back

y(x)=c1cos(aex)+c2sin(aex)

Verification

restart; 
ode:=diff(diff(y(x),x),x)+diff(y(x),x)+a*exp(-2*x)*y(x)=0; 
ys:=_C1*cos(sqrt(a)*exp(-x))+_C2*sin(sqrt(a)*exp(-x)); 
odetest(y(x)=ys,ode); 
0