(10) Using u(x,t) = (A(p) cos pr + B(p) sin pre-p² dp obtain the integral form solution of Ət² = , 2u მე2 satisfying the initial condition u(x, 0) = f(x), where f(x) = 1

Write the system æ' = e"x 4ty + 9 sin(t), y' = 7 tan(t)y + 9x – 4 cos(t) in the form d = P(t) dt Use prime notation for derivatives and write r and x', etc., instead of x(t), x'(t), or 4

Obtain general solution of ODE y''-y=exp(x) + cos(2x)

Consider the function. y = (2x² + 1)3 (a) Decompose the function in the form y = } {f(u), u} = dy = dx { dy (b) Find as a function of x. dx 12x (2x² + 1)² f(u) and u = g(x). (Use non-identity functions for f(u) and u.)

Use variation of parameters to find a particular solution, given the solutions y₁, y2 of the complementary equation sin(x)y"+(2 sin(x) — cos(x))y'+(sin(x) — cos(x))y - Yp(x) = Y₁ = e, y₂=e* cos(x) е

y" + 6y (y')³=0 y(x) = C₂e¹*-6 C₁y+y³=C₂+x C. (C₂-y)² + x² = C₁ d. y=C₂-In(C₁-x²) e. a. b. y(x) = C₂ + In(cos(2(C₁+. +x)

x'y" + 2x²y" = x + sin(In x) Find the general solution of the Cauchy-Euler equation.

Set up the appropriate form of a particular solution yp of the dif-ferential equation   y''''-y''-2y'+2y =xe^(x)+x^(2)cosx

Find the general solution of dy + y cot(x) = 6 csc'(x). dx a) Oy = -6 sec(x) +C b) Oy = 6 cot(x) csc(x) +C c) Oy = -6 cot(x) csc(x)+C d) Oy= 6 cot(x) csc(x) +C csc(x) e) Oy = -6 cot(x) csc(x) +C csc(x) f) O None of the above.

Show that the function f(t) = 1² – } is a solution of the dif- ferential equation (y')² – 4y = 2.