Answer:
![y = 10cos (7x) - \frac{4}{7}sin ( 7x )](https://tex.z-dn.net/?f=y%20%3D%2010cos%20%287x%29%20-%20%5Cfrac%7B4%7D%7B7%7Dsin%20%28%207x%20%29)
B.
B.
![y = \frac{17}{8}e^4^x - \frac{1}{8}e^-^4^x](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B17%7D%7B8%7De%5E4%5Ex%20-%20%5Cfrac%7B1%7D%7B8%7De%5E-%5E4%5Ex)
Step-by-step explanation:
Question 1:
- We are given a homogeneous second order linear ODE as follows:
![y'' + 49y = 0](https://tex.z-dn.net/?f=y%27%27%20%2B%2049y%20%3D%200)
- A pair of independent functions are given as ( y1 ) and ( y2 ):
![y_1 = cos ( 7x )\\\\y_2 = sin ( 7x )](https://tex.z-dn.net/?f=y_1%20%3D%20cos%20%28%207x%20%29%5C%5C%5C%5Cy_2%20%3D%20sin%20%28%207x%20%29)
- The given ODE is subjected to following initial conditions as follows:
![y ( 0 ) = 10\\\\y ' ( 0 ) = -4](https://tex.z-dn.net/?f=y%20%28%200%20%29%20%3D%2010%5C%5C%5C%5Cy%20%27%20%28%200%20%29%20%3D%20-4)
- 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:
![y = c_1y_1 + c_2y_2](https://tex.z-dn.net/?f=y%20%3D%20c_1y_1%20%2B%20c_2y_2)
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:
![y'_1 = -7sin(7x) , y''_1 = -49cos(7x)\\\\y'_2 = -7cos(7x) , y''_2 = -49sin(7x)\](https://tex.z-dn.net/?f=y%27_1%20%3D%20-7sin%287x%29%20%2C%20y%27%27_1%20%3D%20-49cos%287x%29%5C%5C%5C%5Cy%27_2%20%3D%20-7cos%287x%29%20%2C%20y%27%27_2%20%3D%20-49sin%287x%29%5C)
- 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.
![y''_1 + 49y_1 = 0\\\\-49cos(7x) + 49cos ( 7x ) = 0\\0 = 0\\\\y''_2 + 49y_2 = 0\\\\-49sin(7x) + 49sin ( 7x ) = 0\\0 = 0\\\\](https://tex.z-dn.net/?f=y%27%27_1%20%2B%2049y_1%20%3D%200%5C%5C%5C%5C-49cos%287x%29%20%2B%2049cos%20%28%207x%20%29%20%3D%200%5C%5C0%20%3D%200%5C%5C%5C%5Cy%27%27_2%20%2B%2049y_2%20%3D%200%5C%5C%5C%5C-49sin%287x%29%20%2B%2049sin%20%28%207x%20%29%20%3D%200%5C%5C0%20%3D%200%5C%5C%5C%5C)
- 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:
![y = c_1y_1 + c_2y_2\\\\y = c_1*cos(7x) + c_2*sin(7x)\\](https://tex.z-dn.net/?f=y%20%3D%20c_1y_1%20%2B%20c_2y_2%5C%5C%5C%5Cy%20%3D%20c_1%2Acos%287x%29%20%2B%20c_2%2Asin%287x%29%5C%5C)
- To complete the above solution we need to determine the constants [ c1 and c2 ] using the initial conditions given. Therefore,
![y (0) = c_1cos ( 0 ) + c_2sin ( 0 ) = 10\\\\y'(0) = -7c_1*sin(0) + 7c_2*cos(0) = -4\\\\c_1 ( 1 ) + c_2 ( 0 ) = 10, c_1 = 10\\\\-7c_1(0) + 7c_2( 1 ) = -4 , c_2 = -\frac{4}{7}](https://tex.z-dn.net/?f=y%20%280%29%20%3D%20c_1cos%20%28%200%20%29%20%2B%20c_2sin%20%28%200%20%29%20%3D%2010%5C%5C%5C%5Cy%27%280%29%20%3D%20-7c_1%2Asin%280%29%20%2B%207c_2%2Acos%280%29%20%3D%20-4%5C%5C%5C%5Cc_1%20%28%201%20%29%20%2B%20c_2%20%28%200%20%29%20%3D%2010%2C%20c_1%20%3D%2010%5C%5C%5C%5C-7c_1%280%29%20%2B%207c_2%28%201%20%29%20%3D%20-4%20%2C%20c_2%20%3D%20-%5Cfrac%7B4%7D%7B7%7D)
- 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:
![y'' -16y =0](https://tex.z-dn.net/?f=y%27%27%20-16y%20%3D0)
- A pair of independent functions are given as ( y1 ) and ( y2 ):
![y_1 = e^4^x\\\\y_2 = e^-^4^x](https://tex.z-dn.net/?f=y_1%20%3D%20e%5E4%5Ex%5C%5C%5C%5Cy_2%20%3D%20e%5E-%5E4%5Ex)
- The given ODE is subjected to following initial conditions as follows:
![y( 0 ) = 2\\y'( 0 ) = 9](https://tex.z-dn.net/?f=y%28%200%20%29%20%3D%202%5C%5Cy%27%28%200%20%29%20%3D%209)
- 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:
![y = c_1y_1 + c_2y_2](https://tex.z-dn.net/?f=y%20%3D%20c_1y_1%20%2B%20c_2y_2)
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.
![y''_1 - 16y_1 = 0\\\\16e^4^x - 16e^4^x = 0\\\\0 = 0\\\\y''_2 - 16y_2 = 0\\\\16e^-^4^x - 16e^-^4^x = 0\\\\0 = 0](https://tex.z-dn.net/?f=y%27%27_1%20-%2016y_1%20%3D%200%5C%5C%5C%5C16e%5E4%5Ex%20-%2016e%5E4%5Ex%20%3D%200%5C%5C%5C%5C0%20%3D%200%5C%5C%5C%5Cy%27%27_2%20-%2016y_2%20%3D%200%5C%5C%5C%5C16e%5E-%5E4%5Ex%20-%2016e%5E-%5E4%5Ex%20%3D%200%5C%5C%5C%5C0%20%3D%200)
- 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:
![y = c_1y_1 + c_2y_2\\\\y = c_1*e^4^x + c_2*e^-^4^x](https://tex.z-dn.net/?f=y%20%3D%20c_1y_1%20%2B%20c_2y_2%5C%5C%5C%5Cy%20%3D%20c_1%2Ae%5E4%5Ex%20%2B%20c_2%2Ae%5E-%5E4%5Ex)
- To complete the above solution we need to determine the constants [ c1 and c2 ] using the initial conditions given. Therefore,
![y ( 0 ) = c_1 * e^0 + c_2 * e^0 = 2\\\\y' ( 0 ) = 4 c_1 * e^0 - 4c_2 * e^0 = 9\\\\c_1 + c_2 = 2 , 4c_1 - 4c_2 = 9\\\\c_1 = \frac{17}{8} , c_2 = -\frac{1}{8}](https://tex.z-dn.net/?f=y%20%28%200%20%29%20%3D%20c_1%20%2A%20e%5E0%20%2B%20c_2%20%2A%20e%5E0%20%3D%202%5C%5C%5C%5Cy%27%20%28%200%20%29%20%3D%204%20c_1%20%2A%20e%5E0%20-%204c_2%20%2A%20e%5E0%20%3D%209%5C%5C%5C%5Cc_1%20%2B%20c_2%20%3D%202%20%2C%204c_1%20-%204c_2%20%3D%209%5C%5C%5C%5Cc_1%20%3D%20%5Cfrac%7B17%7D%7B8%7D%20%2C%20c_2%20%3D%20-%5Cfrac%7B1%7D%7B8%7D)
- Now we can write the complete solution to the given homogeneous second order linear ODE as follows:
.... Answer