y%3Dsin2x.jpeg "(d^2-3d+2)y=sin2x")
Differential Equation: (d^2-3d+2)y=sin2x
In this article, we will solve the differential equation (d^2-3d+2)y=sin2x. This is a second-order linear differential equation, where d/dx is the differential operator.
General Form of the Solution
The general form of the solution to a second-order linear differential equation is:
y(x) = yc(x) + yp(x)
where yc(x) is the complementary function (the general solution to the homogeneous equation) and yp(x) is the particular integral (a particular solution to the inhomogeneous equation).
Complementary Function (yc)
To find the complementary function, we need to solve the homogeneous equation:
(d^2-3d+2)y=0
This is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation is:
r^2 - 3r + 2 = 0
Solving for r, we get:
r = 1 or r = 2
Therefore, the complementary function is:
yc(x) = Ae^x + Be^2x
where A and B are arbitrary constants.
Particular Integral (yp)
To find the particular integral, we need to find a particular solution to the inhomogeneous equation:
(d^2-3d+2)y=sin2x
Using the method of undetermined coefficients, we assume a particular solution of the form:
yp(x) = Csin2x + Dcos2x
Substituting this into the inhomogeneous equation, we get:
(-4C - 6D + 2C)sin2x + (-4D + 6C + 2D)cos2x = sin2x
Equating the coefficients of sin2x and cos2x, we get:
-4C - 6D + 2C = 1 -4D + 6C + 2D = 0
Solving this system of equations, we get:
C = -1/10 D = 3/20
Therefore, the particular integral is:
yp(x) = (-1/10)sin2x + (3/20)cos2x
General Solution
The general solution to the differential equation is:
y(x) = yc(x) + yp(x) = Ae^x + Be^2x - (1/10)sin2x + (3/20)cos2x
where A and B are arbitrary constants.
Conclusion
In this article, we have solved the differential equation (d^2-3d+2)y=sin2x using the method of undetermined coefficients. The general solution consists of a complementary function and a particular integral. The complementary function is a linear combination of exponentials, while the particular integral is a linear combination of sine and cosine functions.
