Discuss fourier transform and partial differential equation

Question

Assignment:

ODE:

1. Solve y' =  sin (x + y) .

2. Find the complete solution of the ODE y⁽⁴⁾ - y⁽²⁾ - 2y = 12x cos x.

3. Find the complete solution of the ODE y⁽⁴⁾ - y = 6 sin x.

4. Find a second order ODE whose solution is a family of circle with arbitrary radius and center on the x-axis, i.e., (x-a)² + y² = b² where a and b are arbitrary constants.

Fourier series, Fourier Transform and Partial differential equation

5. Write the Fourier series for f(t)= |cost| .

6. Find the Fourier series of a periodic signal with f (t)= exp (|-t|), -1

7. Find the (two - side) Fourier transform F(W) F {ƒ (t)}  of ƒ (t) = t exp(-|t|).

8. Find the Fourier transform X (ƒ) of x(t)= exp (|-t|)cos(2Πƒ_ct).

9. Solve the partial differential equation xz_x + zt = xt for z (x,t), x≥ 0, t≥0 with the condition z (x,0) = 0 and z (0,t) = 0 Hint : Use Laplace transform

10. Solve for z(x t,) the partial differential equation z_xx = z_t + z, t ≥ 0, 0 ≤ x ≤ 1 with the conditions z_x(0,t) = z_x (0,t) = z_x (1,t) = 0 for all t all and z (x,0)= 2sin² Πx for all x.

Laplace and Inverse Laplace Transform

11. Find the laplace transform of ƒ (t) = e^-2t |sin t|u(t).

12. Find the inverse laplace transform of F (S) = 1-2e^-2s + e-^4s/ s²

13. Find the inverse laplace transform of F (S) = 4(s² + 2)/ (s+1)(s² + 1)²

Eigenvalue and Eigenvector

14. Find the eigenvalue and eigenvector of A = [1   1  -2]  and  B= [1  1  1]
                                                                [-1 2   1]             [1  1  1]
                                                                [0  1  -1]             [1  1  1]

Vector space, Basis, Dimensions

15. Find condition on a, b, c so that (a,b,c) ∈ R³ belongs to the space generated by u = (2,1,0), v = (1,-1,2) , and w= (0,3,-4).

16. Let W be the subspace of R⁴ generated by the vectors (1,-2,5,-3), (2,3,1,-4) and (3,8,-3,-5)

a. Find a basis and the dimension of W .

b. Extend the basis of W to a basis of the whole space R⁴.

17. Let U and W be subspaces of R⁵ such that

U is spanned by {(1,3, -3, -1, -4) , (1,4, -1, -2, -2) , (2,9,0, -5, -2)}

W is spanned by {(1,6,2, -2,3) , (2,8, -1, -6, -5) , (1,3, -1, -5, -6)}

a. Find the basis of (U ∩ W ).

b. Find dim (U + W) and dim (U ∩ W).

Residues

18. Evaluate ₀^2Π∫ sin2 Θ dΘ/ 5 + 3cos Θ.

19. Evaluate _c? 1/ z³(z² + 2z +2) where c is the counter - clockwise.

20. Evaluate _-∞^∞∫ 2z²-1 dz/ z³ - z² - 4z - 6

System of linear equation

21. Find the value of so that the solution of the following equations exists. By using that value of solve those equations.

x₁ - x₂ + 2x₃ = 3

-4x₁ + x₂ + 7x₃ =-5

-2x₁ - 3x₂ + 11x₃ = k