y%3Dx-cos2x.jpeg "(d^2-5d+6)y=x Cos2x")
Solving the Differential Equation (d^2-5d+6)y=x cos(2x)
Introduction
In this article, we will solve the differential equation (d^2-5d+6)y=x cos(2x), where y is a function of x. This equation is a second-order linear ordinary differential equation (ODE) with a non-homogeneous term. We will use the method of undetermined coefficients to find the general solution of the equation.
Homogeneous Equation
First, let's consider the homogeneous equation associated with the given differential equation:
(d^2-5d+6)y=0
To solve this equation, we can use the characteristic equation:
r^2-5r+6=0
Factoring the quadratic, we get:
(r-2)(r-3)=0
This gives us two distinct real roots: r=2 and r=3. Therefore, the general solution of the homogeneous equation is:
y_c(x) = c1e^(2x) + c2e^(3x)
where c1 and c2 are arbitrary constants.
Particular Solution
Now, let's find the particular solution of the non-homogeneous equation (d^2-5d+6)y=x cos(2x). We can use the method of undetermined coefficients to find the particular solution.
Assume that the particular solution has the form:
y_p(x) = A cos(2x) + B sin(2x)
where A and B are unknown constants.
Substituting this expression into the non-homogeneous equation, we get:
(d^2-5d+6)(A cos(2x) + B sin(2x)) = x cos(2x)
Expanding and simplifying the equation, we get:
(-4A+4B-5A) cos(2x) + (-4B-5B) sin(2x) = x cos(2x)
Equating the coefficients of cos(2x) and sin(2x), we get:
-4A+4B-5A=0 and -4B-5B=0
Solving these equations, we get:
A=0 and B=-1/17
Therefore, the particular solution is:
y_p(x) = (-1/17) sin(2x)
General Solution
The general solution of the differential equation (d^2-5d+6)y=x cos(2x) is the sum of the homogeneous solution and the particular solution:
y(x) = y_c(x) + y_p(x)
y(x) = c1e^(2x) + c2e^(3x) + (-1/17) sin(2x)
This is the general solution of the differential equation.
Conclusion
In this article, we have solved the differential equation (d^2-5d+6)y=x cos(2x) using the method of undetermined coefficients. The general solution of the equation involves the sum of the homogeneous solution and the particular solution.
