y%3Dsin3x.jpeg "(d2-3d+2)y=sin3x")
Solving the Equation (d2-3d+2)y=sin3x
In this article, we will explore the solution to the differential equation (d2-3d+2)y=sin3x. This is a second-order linear ordinary differential equation (ODE) with a sinusoidal forcing function.
Background
Before we dive into the solution, let's review some background concepts:
- Differential Equations: A differential equation is a mathematical equation that involves an unknown function and its derivatives.
- Linear ODEs: A linear ODE is a type of differential equation where the derivatives of the unknown function are proportional to the function itself.
Solution Method
To solve the differential equation (d2-3d+2)y=sin3x, we will use the method of undetermined coefficients. This method involves assuming a particular solution of the form:
y_p = A sin3x + B cos3x
where A and B are constants to be determined.
Finding the Particular Solution
To find the particular solution, we will substitute y_p into the original differential equation:
(d2-3d+2)(A sin3x + B cos3x) = sin3x
Expanding the left-hand side and collecting terms, we get:
(-9A + 3B) sin3x + (3A + 9B) cos3x = sin3x
Equating coefficients, we have:
-9A + 3B = 1 ... (1)
3A + 9B = 0 ... (2)
Solving the system of equations (1) and (2), we get:
A = -1/10 and B = 1/30
Therefore, the particular solution is:
y_p = (-1/10) sin3x + (1/30) cos3x
Finding the General Solution
To find the general solution, we need to find the complementary function y_c. The complementary function is a solution to the homogeneous equation:
(d2-3d+2)y = 0
The characteristic equation is:
r^2 - 3r + 2 = 0
Solving the characteristic equation, we get:
r = 1 and r = 2
Therefore, the complementary function is:
y_c = C1 e^x + C2 e^(2x)
where C1 and C2 are arbitrary constants.
General Solution
The general solution to the differential equation (d2-3d+2)y=sin3x is:
y = y_p + y_c
y = (-1/10) sin3x + (1/30) cos3x + C1 e^x + C2 e^(2x)
where C1 and C2 are arbitrary constants.
Conclusion
In this article, we have solved the differential equation (d2-3d+2)y=sin3x using the method of undetermined coefficients. The general solution involves a particular solution and a complementary function. The particular solution is a sinusoidal function that satisfies the original differential equation, while the complementary function is a solution to the homogeneous equation.
