Why is general solution of homogeneous equation linear

[lonewolf219]

Hi, I don't understand why the general solution of 2nd order homogeneous equation is linear? Why is c_1e^(xt)+c_2e^(xt) a linear differential equation? What am I missing here? Any help would be appreciated, I'm struggling a bit understanding the concepts of differential equations...

 

[matematikawan]

Hi, I don't understand why the general solution of 2nd order homogeneous equation is linear? Why is c_1e^(xt)+c_2e^(xt) a linear differential equation? What am I missing here? Any help would be appreciated, I'm struggling a bit understanding the concepts of differential equations...

The general solution is a linear combination of two linearly independent solutions y₁(x) and y₂(x).


p.s. I think something is not right with your notation e^(xt) .

 

[lonewolf219]

Yes, you're right. The x in the exponent of e should be r, where you would find the roots using the characteristic equation. C_1e^(r_1t)+C_2e^(r_2t). So this is a linear solution because y_1 and y_2 are to the first power? Even though the function e^rt is not a linear function?

 

[lurflurf]

It is linear in y not x.
L is a linear operator if
L[Ʃa_ny_n]=Ʃa_nL[y_n]

 

[lonewolf219]

Ah, OK. Thanks guys.