Method of Undetermined Coefficient Calculator

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Example

Created on 2024-06-20Asked by Abigail Allen (Solvelet student)

Solve the differential equation using the method of undetermined coefficients:

y'' + 3y' + 2y = e^x.

Solution

To solve the differential equation

y'' + 3y' + 2y = e^x

using the method of undetermined coefficients, follow these steps: 1. Solve the corresponding homogeneous equation:

y'' + 3y' + 2y = 0.

2. Find the characteristic equation:

r^2 + 3r + 2 = 0.

3. Factor the characteristic equation:

(r + 1)(r + 2) = 0.

4. Solve for the roots:

r = -1, \quad r = -2.

5. Write the general solution of the homogeneous equation:

y_h = C_1 e^{-x} + C_2 e^{-2x}.

6. Determine the form of the particular solution

y_p

: Since the right-hand side

e^x

is an exponential function not present in the homogeneous solution, assume:

y_p = A e^x.

7. Substitute

y_p

into the non-homogeneous equation:

(A e^x)'' + 3(A e^x)' + 2(A e^x) = e^x.

8. Simplify and solve for

A

A e^x + 3A e^x + 2A e^x = e^x,

6A e^x = e^x,

A = \frac{1}{6}.

9. Write the general solution:

y = y_h + y_p = C_1 e^{-x} + C_2 e^{-2x} + \frac{1}{6} e^x.

Therefore, the solution to the differential equation is:

y = C_1 e^{-x} + C_2 e^{-2x} + \frac{1}{6} e^x.

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Aiden Campbell on Solvelet

1. Find the particular solution of the nonhomogeneous differential equation

y'' + 2y' + y = e^{2x}

using the method of undetermined coefficients.2. Determine the particular solution of the nonhomogeneous differential equation

y'' - y = x^2 + 2x + 3

using the method of undetermined coefficients.

DefinitionLinear differential equation is an equation of the form y n dif where y n and y n-1 are the linearly dependent functions of a single variable and a x is a differentiable function. Example: are the linear transformations from vector space to vector space. A linear transformation from vector space to vector space or vector space to vector space a linear map of m into n. If there is a linear transformation a: be vectors in and a c is scalar x=c v 1 +c v 2 and ax=c ax. Examples of linear transformations: If T is a linear transformation and L p → L p is defined by. The method of undetermined coefficients

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