Find the general solution of the differential equation

Please find the general solution of the differential equation below. It needs to be converted to standard format (s² + bs + c = 0), the roots found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. Please show all details in the solution. Thank you 2d²y/dx² + 4dy/dx + 5y = 0

Find the Particular Solution satisfying the given initial conditions

Please find the particular solution of the differential equation below satisfying the given initial conditions. The roots need to be found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. After finding the general solution, solve for the values of C given the initial conditions. Then rewrite the equation with those values. Please show all details in the solution. Thank you d²y/dx² + 16y = 0 y(0)=3, y´(0)=12

Find the particular solution of the differential equation

Please find the particular solution of the differential equation below satisfying the given initial conditions. The equation needs to be put in standard form (s² + bs + c = 0). The roots need to be found, and then plugged into the appropriate general formula that satisfies whether b² + 4c is less than, greater than, or equal to zero. After finding the general solution, solve for the values of C given the initial conditions. Then rewrite the equation with those values. Please show all details in the solution. Thank you 4d²y/dx² -4dy/dx + y = 0 y(0)=0, y´(0)=1

Find the particular solution of the differential equation

Find the particular solution of the general equation below, given the initial conditions. Please show all details in the solution, simplifying if possible. Thank you. y=C(e^(-2x))sinx + C(e^(-2x))cos x y(0)= 1 y'(0)= 0

Nonhomogenous 2nd order linear differential equations

Please find the general solution of the nonhomogenous second order linear differential equation below by following these steps: 1. Find the general solution y= C1y1 + C2y2 of the associated homogenous equation (complementary solution) 2. Find a single solution of yp of above.(particular solution). 3. Express the general solution in the form y=yp + C1y1 +C2y2 Please show all steps. Thank you. d²y/dx² - 4y = e^(3x)