Why is general solution of homogeneous equation linear

  • #1
186
2
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...
 

Answers and Replies

  • #2
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) .
 
  • #3
186
2
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?
 
  • #4
lurflurf
Homework Helper
2,440
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It is linear in y not x.
L is a linear operator if
L[Ʃanyn]=ƩanL[yn]
 
  • #5
186
2
Ah, OK. Thanks guys.
 

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