A homogeneous second-order linear differential equation, two functions y 1y1 and y 2y2, and a pair of initial conditions are given. First verify that y 1y1 and y 2y2 are solutions of the differential equation. Then find a particular solution of the form y = c1y1 + c2y2 that satisfies the given initial conditions. Primes denote derivatives with respect to x. y'' + 49y = 0; y1 = cos(7x) y2 = sin(7x); y(0) = 10 y(0)=-4
1.Why is the function y, = e * a solution to the differential equation?
A. The function y1 =e 4X is a solution because when the function and its indefinite integral, , are substituted into the equation, the result is a true statement.
B. The function y1 = e 4X is a solution because when the function and its second derivative, y1" = 16 e 4x, are substituted into the equation, the result is a true statement.
2. Why is the function y2 solution the differential equation?
A. The function y2 = e 4x is a solution because when the function and its indefinite integral, are substituted into the equation, the result a true statement. The function y2 = e 4X is a solution because when the function and its second derivative, y2" = 16 e -4x are substituted into the equation, the result is a true statement. The particular solution of the form y = c, y, +c, y2 that satisfies the initial conditions y(0) 2 and y'(0) = 9 is y =.

B.

B.

Step-by-step explanation:

Question 1:

- We are given a homogeneous second order linear ODE as follows:

- A pair of independent functions are given as ( y1 ) and ( y2 ):

- The given ODE is subjected to following initial conditions as follows:

- We are to verify that the given independent functions ( y1 ) and ( y2 ) are indeed the solution to the given ODE. If the functions are solutions then find the complete solution of the homogeneous ODE of the form:

Solution:-

- To verify the functions are indeed the solution to the given ODE. We will plug the respective derivatives of each function [ y1 and y2 ] into the ODE and prove whether the equality holds true or not.

- Formulate the second derivatives of both functions y1 and y2 as follows:

- Now plug the second derivatives of each function and the functions itself into the given ODE and verify whether the equality holds true or not.

- We see that both functions [ y1 and y2 ] holds true as the solution to the given homogeneous second order linear ODE. Hence, are the solution to given ODE.

- The complete solution to a homogeneous ODE is given in the form as follows:

- To complete the above solution we need to determine the constants [ c1 and c2 ] using the initial conditions given. Therefore,

- Now we can write the complete solution to the given homogeneous second order linear ODE as follows:

                               .... Answer

Question 2

- We are given a homogeneous second order linear ODE as follows:

- A pair of independent functions are given as ( y1 ) and ( y2 ):

- The given ODE is subjected to following initial conditions as follows:

- We are to verify that the given independent functions ( y1 ) and ( y2 ) are indeed the solution to the given ODE. If the functions are solutions then find the complete solution of the homogeneous ODE of the form:

Solution:-

- To verify the functions are indeed the solution to the given ODE. We will plug the respective derivatives of each function [ y1 and y2 ] into the ODE and prove whether the equality holds true or not.

- Formulate the second derivatives of both functions y1 and y2 as follows:

- Now substitute the second derivatives of each function and the functions itself into the given ODE and verify whether the equality holds true or not.

- We see that both functions [ y1 and y2 ] holds true as the solution to the given homogeneous second order linear ODE. Hence, are the solution to given ODE.

- The complete solution to a homogeneous ODE is given in the form as follows:

- To complete the above solution we need to determine the constants [ c1 and c2 ] using the initial conditions given. Therefore,

- Now we can write the complete solution to the given homogeneous second order linear ODE as follows:

                              .... Answer

answer: slight chance that it will happen

step-by-step explanation: