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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Fourier series, Fourier Transform and Partial Differential Equations are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
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