WW 2: Problem 9 (2 points) The differential equation y”

WW 2: Problem 9 (2 points) The differential equation y'' - y = 0 has one of the following two parameter families as its general solution y = c_1e^x + c_2e^{-x} y = c_1 \cos(x) + c_2 \sin(x) y = c_1 \tan(x) + c_2 \sec(x) y = c_1 + c_2x Find the solution such that y(0) = 6 and y'(0) = 4:

Transcript

00:01 In the first question, it is given that y is equal to c1 erase to x plus c2 erase to negative x is the solution of the differential equation y double dash minus y is equal to 0 in the interval negative infinity.

00:22 We are asked to find the solution of the initial value problem y double dash minus y is equal to zero with initial condition y of 0 is equal to 0.

00:32 0 y dash of 0 is equal to 1 which is a member of this family to determine the solution of this initial value problem use these initial conditions and this solution or the general solution first find y dash of x which is the first derivative of y of x it is c1 erased to x minus c2 erase to negative x now use the function y of x is equal to c1 erased to x plus c2 erase to negative x and the initial condition y of 0 is equal to 0 to get 0 is equal to c1 erase to negative 0 that is we get 0 is equal to c1 plus c2 plus c2 this is the first equation marked as equation 1 now use the function y dash of x is equal to c1 erase to x minus c2 erase to negative x and the initial condition y dash of 0 is equal to 1 to get 1 is equal to c1 erase to 0 minus c2 erase to negative 0 from which we get another equation 1 is equal to c1 minus c2 mark us equation 2 now add equation 1 and 2 together to get this as 1 is equal to 2 c1 that is we get c1 is equal to half.

02:06 From the first equation, substituting the value of c1, we get the value of c2 to be negative half.

02:14 Therefore, the solution of the initial value problem is y is equal to half e raised to x minus half e raised to negative x and this is the required member of the family that is the solution of the initial value problem.

02:32 In the second question, it is given that y is equal to c1 erased to 4x plus c2 erase to negative x in the interval negative infinity infinity is the solution of the differential equation y double dash minus three y dash minus four y is equal to zero we are asked to find the solution of the initial value problem y double dash minus three y dash minus four is equal to zero with the initial condition y of zero is equal to one and y dash of zero is equal to 2.

03:05 Again using the general solution we have y dash of x is equal to 4 c1 erase to 4x minus c2 erase to negative x.

03:17 Now using the initial condition y of 0 is equal to 1 and the function y of x is equal to c1 erase to 4x plus c2 erase to negative x we have 1 is equal to c1 plus c2 and using this derivative function and the initial condition, y of y dash of 0 is equal to 2, we have 2 is equal to 4c1 minus c2.

03:45 Now adding these together we get 3 is equal to 5c1...

Question

WW 2: Problem 9 (2 points) The differential equation y'' - y = 0 has one of the following two parameter families as its general solution y = c_1e^x + c_2e^{-x} y = c_1 \cos(x) + c_2 \sin(x) y = c_1 \tan(x) + c_2 \sec(x) y = c_1 + c_2x Find the solution such that y(0) = 6 and y'(0) = 4:

Please give Ace some feedback