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The following equations are for your reference: / sin(bx)dx sin(br) b cos(br)] +C / a2 bsin(bx)++ C 2ap C{t sin(at)} (²+a²)² C(tood(at)} ²-a² (p=++2)F (-1)" sinz ,2m+1 == (2n+1)! =o (2n)! Q1: We have f(t) Compute the Laplace transform of f(t) using the definition of Laplace transform Q2: We have where a >0andu>>0 are parameters. Write down the Laplace transform of f(t). Q3: Write down (i) the Laplace transform of sint; (ii) the Laplace transform of sint: (iii) the Laplace transform of te sint. Q4: Find C-1 (ii) (For non-300 level students only) Find C-1 Q6: Using Laplace transform to solve the ODE dx d²z dx subject to the initial conditions Q7 Given function f(r), what isthe definition of its Fourier transform? Q8: Prove where parameter. Q9: Use the definition of Fourier transform to prove that Q10: Assurning that lime +00 u(z) =0 and limig da da = 0, use the Fourier transform method to find the solution of the ODE: d²u dx² 2 Q1 Solve the differential equation Ly==dy-2y=0 by assuming solution of the form A1x A2x² -AzI3 and assuming that the series converges in an interval including 0. Q12: Use the Frobenius method tofind the general solution for by assuming solution of the form y

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Differential Equations
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