Solution steps plz 5. Use Duhamel's formula to express the solution of the {x"-7x'+12x = f(t)...
Solution steps plz k=0 x"+x=Ž8(t-2k1) 1. Solve the IVP x(0)=0, x'(0) = 0 and discuss the behavior of the oscillator's amplitude when t → 00.
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but we would be left with an expression in terms of sin(7x) with no extra cos(7x) factor. Instead, we separate a single sine factor and rewrite the remaining sin" (7x) factor in terms of cos(7x): sin'(7x) cos”(7x) = (sinº(7x))2 cos(7x) sin(7x) = (1 - Cos?(7x))2 cos?(7x) sin(7x). in (7x) cos?(7x) and ich is which? Substituting u = cos(7x), we have du = -sin (3x) X...
Solution steps plz 2. Solve the IVP (x" + x = 3uſt-27) |x(O)=0, x'(O)= 2
9. Use a suitable Fourier Transform to find the solution of the IVP utt (x, t) Uz(0, t) u(x, t) , uz (z, t) 4uzz (x, t) + q (x, t), 0, t> 0, 0as x → 00, x > 0, t > 0, = = t>0. → = 0, ut (2,0)-( = { t, 0 0-x-2, -1, 0, > 2, u(x, 0) q(a, t) Leave your answer in the form of an integral. 9. Use a suitable Fourier Transform...
plz show steps thx 5. a) Evaluate st -2x + 7x? dx. b) Evaluate: [** sin(x* +1]dx. c) Evaluate: [(3x? –5x+4-4e”)dx
5. Express f(t) using the unit step function an then use the Laplace Transform to solve the given IVP: y' + y = f(t), y(0) = 0, where f(t) = So, ost<1 15, t21
Let f(x, y) = 7x²y + 2x + 2. Evaluate f(5,5). (Express numbers in exact form. Use symbolic notation and fractions where needed.) f(5,5) = Evaluate f(x + d, y). (Express numbers in exact form. Use symbolic notation and fractions where needed.) f(x + d, y) = Evaluate f(x, y + d). (Express numbers in exact form. Use symbolic notation and fractions where needed.) f(x, y + d) =
Please write neatly Let f(x) be a Cl(R) function and u(t, x) be a solution to the inviscid Burger's IVP Ut + uuz = 0, 4(0,x) = f(x). (a) Use the formula ult, x) = f (x – ut) to derive the following explicit identity for the partial derivative Ou du Ət f($)f'(5) 1+tf'(E) with $ = X – ut. (b) Assume that there exists a number wo E R such that f'(20) < 0 and f(x) > 0. Use...
10. Use Duhamel's principle to find a bounded solution to utAu+ f(r,t), 0<r< R, t 0, u(R,t) 0, t>0, u(r,0) 0, 0sr <R. 10. Use Duhamel's principle to find a bounded solution to utAu+ f(r,t), 0
Please show all steps to solution. 7. Use a suitable Fourier Transform to find the solution of the IVP 2t-r-1 ,2-1 t 〉 0, , u(x, t), uz (x, t) 0asx→00, t〉0, → 7. Use a suitable Fourier Transform to find the solution of the IVP 2t-r-1 ,2-1 t 〉 0, , u(x, t), uz (x, t) 0asx→00, t〉0, →