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Use the method of undetermined coefficients to solve the following differential equation y′′−2y′+y=sinx

User Shenkwen
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The general solution is y = y_h + y_p, where y_h is the general solution to the homogeneous equation.

The method of undetermined coefficients is a technique used to find a particular solution to a nonhomogeneous linear differential equation. In this case, we have the differential equation:

y'' - 2y' + y = sin(x)

To use the method of undetermined coefficients, we assume that the particular solution can be written as a linear combination of functions that are similar to the nonhomogeneous term, in this case, sin(x). Since sin(x) is a trigonometric function, we assume that the particular solution is of the form:

y_p = A sin(x) + B cos(x)

where A and B are constants that we need to determine.

Now, we find the first and second derivatives of y_p:

y'_p = A cos(x) - B sin(x)

y''_p = -A sin(x) - B cos(x)

Substituting these derivatives into the original differential equation, we get:

(-A sin(x) - B cos(x)) - 2(A cos(x) - B sin(x)) + (A sin(x) + B cos(x)) = sin(x)

Simplifying, we get:

-2A cos(x) + 2B sin(x) = 0

Since this equation must hold for all x, the coefficients of cos(x) and sin(x) must individually equal zero:

-2A = 0 (1)

2B = 0 (2)

From equation (2), we can solve for B and find that B = 0.

Substituting B = 0 into equation (1), we find that -2A = 0, which implies that A = 0.

Therefore, the particular solution y_p is:

y_p = A sin(x) + B cos(x) = 0

So, the solution to the given differential equation is the sum of the general solution to the homogeneous equation (which we haven't found) and the particular solution:

y = y_h + y_p

where y_h is the general solution to the homogeneous equation.

User Demyn
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