y%3De%5E3x.jpeg "(d^2-3d+2)y=e^3x")
Solving the Differential Equation (d^2-3d+2)y=e^3x
In this article, we will solve the differential equation (d^2-3d+2)y=e^3x. This is a second-order linear differential equation, and we will use the method of undetermined coefficients to find the general solution.
Step 1: Find the Complementary Solution
The complementary solution is the solution to the homogeneous equation (d^2-3d+2)y=0. To find the complementary solution, we can use the characteristic equation:
r^2 - 3r + 2 = 0
factoring the quadratic equation, we get:
(r-1)(r-2) = 0
which gives us two distinct roots: r=1 and r=2.
The complementary solution is therefore:
y_c = c1e^x + c2e^(2x)
where c1 and c2 are arbitrary constants.
Step 2: Find the Particular Solution
To find the particular solution, we need to find a function y_p that satisfies the nonhomogeneous equation (d^2-3d+2)y=e^3x.
Assuming a particular solution of the form y_p = Ae^(3x), where A is a constant, we can substitute this into the differential equation to get:
(d^2-3d+2)(Ae^(3x)) = e^3x
Expanding and simplifying, we get:
9Ae^(3x) - 9Ae^(3x) + 2Ae^(3x) = e^3x
which simplifies to:
2Ae^(3x) = e^3x
Dividing both sides by e^(3x), we get:
2A = 1
A = 1/2
so the particular solution is:
y_p = (1/2)e^(3x)
Step 3: Find the General Solution
The general solution is the sum of the complementary solution and the particular solution:
y = y_c + y_p
y = c1e^x + c2e^(2x) + (1/2)e^(3x)
This is the general solution to the differential equation (d^2-3d+2)y=e^3x.
Conclusion
In this article, we have solved the differential equation (d^2-3d+2)y=e^3x using the method of undetermined coefficients. We found the complementary solution and the particular solution, and combined them to get the general solution. The general solution is y = c1e^x + c2e^(2x) + (1/2)e^(3x), where c1 and c2 are arbitrary constants.
