Question

y'′+4y=−12sin2x,y(0)=1.8,y′(0)=5.0 Exercise 9 Find a (real) general solution. Show each step of your work. y′′+10y +25y=e^−5x

Answer 1

The question involves solving second order linear differential equations using methods like undetermined coefficients or variation of parameters.

Step-by-step explanation:

The student's question revolves around solving a second order linear differential equation. The particular equation given is y'' + 4y = -12sin(2x), which can be solved using the method of undetermined coefficients. Additionally, the student is instructed to find the general solution for a modified equation y'' + 10y + 25y = e^{-5x}.

This too is a second order linear differential equation, and it would typically be solved using the method of variation of parameters or undetermined coefficients, though without the full context of the question some steps cannot be provided.

Remember, when solving these equations, to look for both homogeneous and particular solutions and then use the initial conditions to find any arbitrary constants in the solution.

The complete question is: Solve the initial value problem y′′+4y=−12sin2x, subject to y(0)=1.8 and y′(0)=5.0

Answer 2

To find the general solution to the differential equation y'' + 4y = -12sin(2x), we solve the homogeneous equation y'' + 4y = 0 and find the general solution y = c1cos(2x) + c2sin(2x). By using the method of undetermined coefficients, we find a particular solution and add it to the general solution of the homogeneous equation.

Step-by-step explanation:

To find the general solution to the differential equation y'' + 4y = -12sin(2x), we can first find the complementary solution by solving y'' + 4y = 0. The characteristic equation is r^2 + 4 = 0, which gives us r = ±2i. Therefore, the general solution to the homogeneous equation is y = c1cos(2x) + c2sin(2x).

Next, we can find a particular solution to the nonhomogeneous equation. Since the right side is -12sin(2x), we can guess a particular solution in the form y = Asin(2x). Plugging this into the equation, we get -4Asin(2x) + 4Asin(2x) = -12sin(2x). This yields 0 = -12sin(2x), which is not true for all values of x. Therefore, this guess does not work.

In order to find a particular solution, we can use the method of undetermined coefficients. Since the right side of the equation is -12sin(2x), we can guess a particular solution in the form y = a + bsin(2x) + ccos(2x). Plugging this into the equation, we get -4b - 4asin(2x) + 4c + 4bcos(2x) = -12sin(2x). By matching the coefficients on both sides, we can solve for a, b, and c. Once we have the particular solution, we can add it to the general solution of the homogeneous equation to get the general solution to the nonhomogeneous equation.