Solving for Particular Solution in a Differential Equation

In summary, the concept of "Undetermined coefficients" is a technique used in solving differential equations by expressing the solution as a linear combination of simpler functions with unknown coefficients. It is particularly useful for non-homogeneous linear equations with constant coefficients and is more efficient compared to other methods. Its key assumptions include linearity, constant coefficients, and non-zero right-hand side, while its limitations include only being applicable to certain types of equations. The method can be extended to higher-order equations by using the principle of superposition and the number of undetermined coefficients needed depends on the order of the equation and any repeated roots.
  • #1
rbailey5
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Homework Statement


determine the particular solution for the differential equation 2x^double prime+x=3t^2


Homework Equations





The Attempt at a Solution


since F(t)=3t^2 I used At^2+Bt+C and the first derivative is 2A+B

plugging back in I get
At^2+(4A+B)t+(2B+C)=3t^2

is this correct?
how do I solve for the variables?
 
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  • #2
I got A=3, B=-12, and C=24 does that look right?
 

Related to Solving for Particular Solution in a Differential Equation

1. What is the concept of "Undetermined coefficients" in mathematics?

The concept of "Undetermined coefficients" is a technique used in solving differential equations. It is based on the idea that a solution to a differential equation can be expressed as a linear combination of simpler functions with unknown coefficients. These coefficients are determined by substituting the solution into the original equation and solving for them.

2. How is the method of "Undetermined coefficients" different from other methods of solving differential equations?

The method of "Undetermined coefficients" is different from other methods, such as the method of variation of parameters, in that it is particularly useful for solving non-homogeneous linear differential equations with constant coefficients. It is also a more straightforward and efficient approach compared to other methods.

3. What are the key assumptions made when using the method of "Undetermined coefficients"?

The key assumptions made when using the method of "Undetermined coefficients" are that the differential equation is linear, has constant coefficients, and has a non-zero right-hand side. Additionally, the method assumes that the solution can be expressed as a linear combination of simpler functions, and that the coefficients of these functions are constants.

4. Are there any limitations to using the method of "Undetermined coefficients"?

While the method of "Undetermined coefficients" is a powerful tool for solving certain types of differential equations, it does have its limitations. It can only be used for linear differential equations with constant coefficients and a non-zero right-hand side. Additionally, it may not work for more complex equations or when there are repeated roots of the characteristic equation.

5. How can the method of "Undetermined coefficients" be extended to solve higher-order differential equations?

The method of "Undetermined coefficients" can be extended to solve higher-order differential equations by using the principle of superposition. This means that the solution can be expressed as the sum of individual solutions, each of which is determined using the same method as for first-order equations. The number of undetermined coefficients needed to solve a higher-order equation depends on the order of the equation and any repeated roots of the characteristic equation.

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