If u(x, t) is the solution of then ди = J²u It მე-2 ა ,0 < x < 1,t > 0, u(x, 0) = 1+x+ sin (πx) cos (πx), u(0, t) = 1, u(1, t) = 2, (a) u() = (b) (ਨੂੰ, ਨੂੰ) = (c) u()+½½-3² = (d) u(1, 1) = e−4 -42

Consider the equation ut = Uxx, 0 0. Suppose u(0, t) = 0, u(1, t) = 0. Suppose u(x, 0) = -3 sin(x) + 2 sin(2x) + 3 sin(3πx) + 4 sin(4x) Fill in the constants in the solution: u(x, t) = help (numbers) е -² sin(x) + e-(2) sin(2x) + -(3π)² sin(3πx) + e e (4) ² sin(4x)

4. Suppose the following functions are a general solution of: y(4) + azy" + a2y" + a1y' + aoy = 0) where a3, a2, 41, aq are constants. Use each general solution to determine the constants a3, a2, a1, a0- (a) y(t) = c1 + czt + c3 Cos(3t) + c4 sin(3t) (b) y(t) = c1 cos(t) + c2 sin(t) + c3 Cos(2t) + c4 sin(2t)

Consider that Let the solution of equation y₁ = x² y2 = 3x² - y3 = 4x-2 2 x² x²y" + xy' 4y = 0 Determine general solution of x²y" + xy' -4y = x³

Verify that y(t)=C_(1)sin5t+C_(2)cos5t is a solution of y''(t)+25y(t)=0

This is the solution of the Cauchy - Euler equation x²y" + 5xy' + 20y = 0 O y = C1x 4 cos(4 In(x)) + C2a 4 sin(4 In(x)) O y = C1a 2 cos(4 In(x)) + C2x 2 sin(4 In(x)) y = C1x 2 cos(2 In(x)) + C2x 2 sin(2 In(x)) O y = C1x 4 cos(3 In(x)) + C2x sin(3 In(x))

The general solution to the ODE x²y" + 2xy' - 2y = 0 is y = C1x + C2x¯2. Use variation of parameters to find a particular solution to x²y" + 2xy' – 2y = x.

A system x'=Ax has the solutions cos (t) (c() a and sin (t) -cos (t) sin (t) Which of the following is also a solution? O a) O O b) O c) O d) -2 sin (t) 2 cos (t) sin (t) + cos (t) 4 sin (t)- 2 cos (4 (t)) (t) 2 sin (t) + 3 cos (t) -sin (t) os (t)) 2 sin (t) + cos (t) 4 sin (t)- 3 cos (t)