Why is general solution of homogeneous equation linear

In summary, the general solution of a 2nd order homogeneous equation is linear because it is a linear combination of two linearly independent solutions. The notation e^(xt) should actually be e^(rt) where r is found using the characteristic equation. The solution is considered linear because the function e^rt is linear in y, not x.
  • #1
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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...
 
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  • #2
lonewolf219 said:
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
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
It is linear in y not x.
L is a linear operator if
L[Ʃanyn]=ƩanL[yn]
 
  • #5
Ah, OK. Thanks guys.
 

1. Why is it important to understand the linearity of general solutions of homogeneous equations?

Understanding the linearity of general solutions of homogeneous equations is important because it allows us to solve for a wide range of variables and constants, making it a versatile tool in many areas of science and mathematics.

2. What does it mean for a solution to be "homogeneous"?

A homogeneous solution is one in which all terms have the same degree or power, meaning there are no constant terms. In other words, the solution is not affected by any changes in the independent variable.

3. How do we determine if a solution to a homogeneous equation is linear?

A solution to a homogeneous equation is linear if it satisfies the properties of linearity, which include additivity and scalability. Additivity means that when two solutions are added together, the resulting solution is also a valid solution. Scalability means that when a solution is multiplied by a constant, the resulting solution is also a valid solution.

4. Can a non-linear equation have a linear general solution?

No, a non-linear equation cannot have a linear general solution. A linear general solution implies that the equation is linear, meaning that it follows the properties of linearity. If an equation is non-linear, it does not follow these properties and therefore cannot have a linear general solution.

5. How do we use the linearity of general solutions to solve for unknown constants?

We can use the linearity of general solutions to solve for unknown constants by plugging in different values for the variables and using the properties of linearity to manipulate the equation and solve for the unknown constants. This method is commonly used in solving systems of linear equations.

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