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Linear Ordinary Differential Equations: Solving (d^2-3d+2)y=sin(e^x)
In this article, we will explore the solution to the linear ordinary differential equation (d^2-3d+2)y=sin(e^x). This type of equation is commonly encountered in various fields of mathematics, physics, and engineering.
The Given Equation
The given equation is:
(d^2-3d+2)y=sin(e^x)
where y is the dependent variable, x is the independent variable, and d is the differential operator.
Understanding the Equation
To solve this equation, we need to understand the properties of the differential operator and the sine function. The differential operator d is defined as:
d = (d/dx)
The sine function sin(e^x) is a periodic function that oscillates between -1 and 1.
Method of Solution
To solve the equation, we will use the method of undetermined coefficients. This method involves assuming a particular solution of the form:
y_p = Ae^x + Be^-x + Ce^(2x) + Dsin(e^x) + Esin(e^x)
where A, B, C, D, and E are constants to be determined.
Substitution
Substitute the particular solution into the given equation:
(d^2-3d+2)(Ae^x + Be^-x + Ce^(2x) + Dsin(e^x) + Esin(e^x)) = sin(e^x)
Simplify the equation by expanding the brackets and collecting like terms:
(-A + 4B - 2C)e^x + (A + 2B - C)e^-x + (2C + D)e^(2x) + (D - E)sin(e^x) = sin(e^x)
Equating Coefficients
Equating the coefficients of like terms on both sides of the equation, we get:
-A + 4B - 2C = 0 ... (1) A + 2B - C = 0 ... (2) 2C + D = 0 ... (3) D - E = 1 ... (4)
Solving the System
Solve the system of equations (1)-(4) to find the values of A, B, C, D, and E.
After some algebraic manipulations, we get:
A = -1 B = 1/2 C = -1/4 D = 1/4 E = -3/4
Particular Solution
Substitute the values of A, B, C, D, and E into the particular solution:
y_p = -e^x + (1/2)e^-x - (1/4)e^(2x) + (1/4)sin(e^x) - (3/4)sin(e^x)
General Solution
The general solution of the equation is the sum of the particular solution and the homogeneous solution:
y = y_p + y_h
where y_h is the homogeneous solution of the equation (d^2-3d+2)y=0.
The homogeneous solution can be found by solving the characteristic equation:
r^2 - 3r + 2 = 0
which gives:
r = 1, 2
Thus, the homogeneous solution is:
y_h = c_1e^x + c_2e^(2x)
where c_1 and c_2 are arbitrary constants.
Final Solution
The final solution of the equation (d^2-3d+2)y=sin(e^x) is:
y = -e^x + (1/2)e^-x - (1/4)e^(2x) + (1/4)sin(e^x) - (3/4)sin(e^x) + c_1e^x + c_2e^(2x)
where c_1 and c_2 are arbitrary constants.
