y%3Dsin3x.jpeg "(d2+4)y=sin3x")
Solving the Equation (d2+4)y = sin(3x)
In this article, we will discuss how to solve the differential equation (d2+4)y = sin(3x). This equation is a second-order linear ordinary differential equation, and we will use the method of undetermined coefficients to find the general solution.
Step 1: Find the Homogeneous Solution
The homogeneous equation is obtained by setting the right-hand side of the equation equal to zero:
(d2+4)y = 0
This is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation is:
r^2 + 4 = 0
Solving for r, we get:
r = ±2i
The general solution to the homogeneous equation is:
y_h = c1 cos(2x) + c2 sin(2x)
Step 2: Find a Particular Solution
To find a particular solution, we will use the method of undetermined coefficients. We will assume that the particular solution has the form:
y_p = A sin(3x) + B cos(3x)
Substituting this into the original equation, we get:
(d2+4)(A sin(3x) + B cos(3x)) = sin(3x)
Expanding and equating coefficients, we get:
-9A + 4A = 1 -9B + 4B = 0
Solving for A and B, we get:
A = -1/5 B = 0
The particular solution is:
y_p = -(1/5) sin(3x)
Step 3: Find the General Solution
The general solution is the sum of the homogeneous solution and the particular solution:
y = y_h + y_p = c1 cos(2x) + c2 sin(2x) - (1/5) sin(3x)
This is the general solution to the differential equation (d2+4)y = sin(3x).
Conclusion
In this article, we have shown how to solve the differential equation (d2+4)y = sin(3x) using the method of undetermined coefficients. The general solution is a linear combination of the homogeneous solution and the particular solution. This equation has many applications in physics and engineering, and its solution is an important tool for modeling real-world systems.
