question 4 please! Find the solution of the ordinary differential equation x log

Question

question 4 please!

Find the solution of the ordinary differential equation x log_e (x) dy/dx + y = x^3 log_e (x) subject to the boundary condition y(e) = 0. Write the final solution in explicit form y = f(x). (a) Use the formal definition of the Laplace Transform to find {e^alpha t cos (omega t)} for constants alpha elementof R and omega > 0. Make sure mathematical rigour is applied, all key steps are clearly explained and give clear reasoning regarding the allowed range for s for which this Laplace transform exists. (b) Evaluate the inverse Laplace transform ^-1 {6/2s - 3 - 3 + 4s/9s^2 - 16} ing and clearly state each Laplace transform property/rule used. Use Laplace transforms to solve the initial value problem d^2 x/dt^2 - 2 dx/dt + 5x = cos (2t), x(0) = 1, dx/dt|_t=0 = 0 ing and clearly state each Laplace transform property/rule used.

Explanation / Answer

4. Just by rearranging, the cos(2t) which will becom c^2 -2^2 and then it will be verified.

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