solve d^3y/dx^3-3d^2y/dx^2+2dy/dx=exp(2x)
y"' - 3y" + 2y' = e^(2x)
D(D-1)(D-2)(y) = e^2x
y = (1/4)(c1)e^x + (1/4)(c2)e^(2x) + (1/2)xe^(2x) + c3
solve d^3y/dx^3+4d^2y/dx^2+4dy/dx=exp(-2x)
that is just like the previous one. Just review your text section on linear DE's. There must be examples just like your problem.
how can find on the particular solution of the linear DE's
d^3y/dx^3+4d^2y/dx^2+4dy/dx=exp(-2x)
To solve the given differential equation:
d^3y/dx^3 - 3d^2y/dx^2 + 2dy/dx = exp(2x)
We can use the method of annihilators.
Step 1: Guess the particular solution (yp) to the given equation by using the form of the right-hand side (exp(2x)).
In this case, since the right-hand side is exp(2x), we can guess a particular solution of the form:
yp = Ae^(2x)
where A is a constant we need to determine.
Step 2: Differentiate the guessed particular solution, yp, to match the highest order of derivatives on the left-hand side of the equation.
dy/dx = 2Ae^(2x)
d^2y/dx^2 = (d/dx)(2Ae^(2x)) = 4Ae^(2x)
d^3y/dx^3 = (d/dx)(4Ae^(2x)) = 8Ae^(2x)
Step 3: Substitute the differentiated particular solution into the given differential equation.
8Ae^(2x) - 3(4Ae^(2x)) + 2(2Ae^(2x)) = exp(2x)
Step 4: Simplify the equation to solve for A.
8Ae^(2x) - 12Ae^(2x) + 4Ae^(2x) = exp(2x)
-4Ae^(2x) = exp(2x)
Step 5: Solve for A.
Divide both sides by -4e^(2x):
A = -exp(2x) / (4e^(2x))
Step 6: Substitute the value of A back into the particular solution, yp.
yp = (-exp(2x) / (4e^(2x))) * e^(2x)
yp = -exp(2x) / 4
So, the particular solution is yp = -exp(2x) / 4.
Step 7: Find the complementary solution.
To find the complementary solution, we solve the homogeneous part of the differential equation:
d^3y/dx^3 - 3d^2y/dx^2 + 2dy/dx = 0
The characteristic equation of this homogeneous part is:
r^3 - 3r^2 + 2r = 0
Factorize the equation:
r(r^2 - 3r + 2) = 0
This gives us three roots:
r1 = 0, r2 = 1, and r3 = 2.
Step 8: Write down the general solution.
The general solution of the homogeneous part is:
y = c1 + c2e^x + c3e^(2x)
where c1, c2, c3 are constants to be determined.
Step 9: Write down the final solution.
The general solution of the entire differential equation is the sum of the particular solution (yp) and the general solution of the homogeneous part (y):
y = yp + c1 + c2e^x + c3e^(2x)
where c1, c2, c3 are arbitrary constants.