# Fourier series, Fourier Transform and Partial Differential Equation

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ODE:

1. Solve ()'sinyxy=+.

2. Find the complete solution of the ODE ()()42212cosyyyx−−=.

3. Find the complete solution of the ODE ()46sinyy−=.

4. Find a second order ODE whose solution is a family of circle with arbitrary radius and center on the x-axis, i.e., ()22xayb−+= where a and b are arbitrary constants.

Fourier series, Fourier Transform and Partial differential equation

5. Write the Fourier series for ()cosftt=.

6. Find the Fourier series of a periodic signal with ()()exp,11ftt=−−<

7. Find the (two - side) Fourier transform ()(){}FWFft= of ()()expfttt=−.

8. Find the Fourier transform ()Xf of ()()()expcos2cxttπ=− .

9. Solve the partial differential equation xtxzzx+= for (),,0,0zxtxt≥≥ with the condition and (),00zx=()0,0zt= Hint : Use Laplace transform

10. Solve for (,zxt the partial differential equation

,0,01xxtzzztx=+≥≤≤

with the conditions ()()0,1,0xxztzt= for all t and ()2,02sinzxxπ= for all x.

Laplace and Inverse Laplace Transform

11. Find the laplace transform of ()()2sintftetut−=.

12. Find the inverse laplace transform of ()24212sseeFSs−−−+=.

13. Find the inverse laplace transform of ()()()()2224211sFSss+=++.

Eigenvalue and Eigenvector

14. Find the eigenvalue and eigenvector of 112121011A−⎡⎤⎢⎥=−⎢⎥⎢⎥−⎣⎦ and . 111111111B⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦

Vector space, Basis, Dimensions

15. Find condition on so that ,,abc()3,,abcR∈belongs to the space generated by and ()()2,1,0,1,1,2,uv==−()0,3,4w=−.

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

a. Find a basis and the dimension of W.

b. Extend the basis of W to a basis of the whole space 4R.

17. Let Uand W be subspaces of 5R 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 ()UW∩.

b. Find dim ()UW+ and dim ()UW∩.

Residues

18. Evaluate 220sin53cosdπθθθ+∫.

19. Evaluate ()32122cdzzzz++∫�� where c is the counter - clockwise.

20. Evaluate 2322146zdzzzz∞−∞−−−−∫.

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. k,k

12312312323472311xxxxxxxxx−+=−++=−−−+=

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