Match the following differential equations with their solutions. The symbols A, B, C in the solutions stand for arbitrary constants. You must get all of the answers correct to receive credit. 1. d^2y/dx^2 + 25y = 0
2. dy/dx = 2xy/x^2 - 5y^2
3. d62y/dx^2 + 16 dy/dx + 64y = 0
4. dy/dx = 10xy
5. dy/dx + 24x^2y = 24 x^2
A. y = Ce^-8x^3 + 1
B. 3yx^2 - 5y^3 = C
C. y = Ae^-8x + Bxe^-8x
D. y = Ae^5x^2
E. y = A cos(5x) + B sin(5x)

Step-by-step explanation:

1.    

    The characteristic equation for the given differential equation is:

   Since the roots are complex

Now, the general solution is:

2.    

  Divide both sides by

  Let,

 Here,

 I.F.

 Now, the general solution is:

3.    

   The characteristic equation is:

 Since the roots are real and repeated.

 Now, the general solution is:

4.    

 Integrating both sides

5.      

      Here,

  I.F.

  Now, the general solution is:

               Let,

h= 12 inches

step-by-step explanation: