Why is the solution in the form of Ce^kx ?

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In summary, the solution to linear differential equations with constant coefficients is sought in the form of Ce^kx because it follows directly from the definition of an exponential function. This approach also utilizes linear algebra and the fundamental theorem of algebra to find a unique solution by satisfying initial conditions and using linearly independent solutions.
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joo
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Why is the solution to linear differential equations with constant coefficients sought in the form of Ce^kx ?

I have heard that there is linear algebra involded here.

Could you please elaborate on this ?
 
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Once you provide initial conditions, those differential equations will define a unique solution (Since in principle you can just numerically integrate it).
So once you have n (the order of the DE) linearly independent solutions (and thus are able to satisfy the initial conditions), you have a unique solution.

This is my understanding, there is probably some more technical way to put it.
 
  • #3
The solution of a first-order linear DE with constant coefficients is an exponential function follows directly from the definition of an exponential function.

For an n'th order DE, either you can convert it into a system of n first-order DE's, or the fundamental theorem of algebra says that you can always factorize it as ##(\frac{d}{dx} - a_1)(\frac{d}{dx} - a_2)\cdots(\frac{d}{dx} - a_n)y(x) = 0##.
 
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1. Why is the solution in the form of Ce^kx?

The solution in the form of Ce^kx is a result of solving differential equations using the method of separation of variables. This method involves separating the dependent and independent variables in the equation and finding a solution in the form of a product of two functions. The exponential function, e^kx, is often found to be one of these functions, and the constant C is included to account for any initial conditions.

2. What is the significance of the constant C in the solution?

The constant C, also known as the arbitrary constant, is an important part of the solution as it accounts for any initial conditions that may exist. In other words, it allows the solution to be adjusted to fit specific boundary conditions or initial values, making it a more accurate representation of the problem at hand.

3. Can the exponential function in the solution be replaced with a different function?

Yes, the exponential function e^kx can be replaced with a different function in certain cases. For example, if the differential equation involves trigonometric functions, the solution may be in the form of a trigonometric function instead of an exponential one. However, in many cases, the exponential function is the most suitable choice for finding a solution.

4. How does the value of k affect the solution?

The value of k, known as the constant of integration, determines the rate of change of the solution. A larger value of k results in a steeper curve, indicating a faster rate of change, while a smaller value of k results in a flatter curve, indicating a slower rate of change. The specific value of k is determined by the initial conditions and the specific differential equation being solved.

5. Is the solution in the form of Ce^kx applicable to all differential equations?

No, the solution in the form of Ce^kx is not applicable to all differential equations. It is a specific method used to solve linear differential equations with constant coefficients. Other types of differential equations may require different methods and have different solutions. It is important to carefully analyze the given equation and choose the appropriate method for solving it.

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