he general solution of the linear system X' = AX is given. 4 cos(4t) sin(4t) ) -4 sin(4t) +C, A = X(t) = e cos(4t) (a) In this case discuss the nature of the solution in a neighborhood of (0, 0). O If X(0) = X, lies on the line y = -4x, then X(t) approaches (0, 0) along this line. Otherwise X(t) approaches (0, 0) from the direction determined by y = -x/4. O If X(0) = X, lies on the line y = -x/4, then X(t) approaches (0, 0) along this line. Otherwise X(t) approaches (0, 0) from the direction determined by y = -4x. O If X(0) = X, lies on the line y = -x/4, then X(t) becomes unbounded along this line. Otherwise X(t) becomes unbounded and y = -4x serves as an asymptote. O If X(0) = X, lies on the line y = -4x, then X(t) becomes unbounded along this line. Otherwise X(t) becomes unbounded and y = -x/4 serves as an asymptote. O All solutions spiral toward (0, 0).

Q: y" + 4y = 2 sin 3x + 3 cos 2x Find the solution y(x)

A: Given differential equation We have to find the solution of the differential equation.

Q: Consider the equation ut = Uxx, 0 0. Suppose u(0, t) = 0, u(1, t) = 0. Suppose u(x, 0) = -3 sin(x) +…

A: The given equation is  ut=uxx ,0<x<1,t>0. Boundary Condition: u(0,t)=u(1,t)=0. Also…

Q: (b) (2, -5π/6) (r, 0) = (r, 8) = y -1 -2 -3 -5 -1.5 (r> 0) ] ) (r< 0) 1 -1.0 2 3 -0.5 4 y -0.2 -0.4…

A:

Q: Write the system y1 = -y1+2 cos(r)y3 + e 2 = xy1 - 10y2 + e" y3 +20 sin(x) Y3 = e2" y2 – a²y3 %3D…

A: The matrix form of given system of differential equations as follows

Q: 1. Let D = the sales of DVDs in billions of dollars, S = the total sales of streaming videos in…

A: The sales of dvds given as the function D = -0.92t + 8.75 and the total sales of streaming videos…

Q: Q.4) The Wronskian W(e~2", e",e²), is: a) 12e b) –12e c) 6e² d) -6e"

A:

Q: (b) y" – 2y' + 5y = eª cos 2x, y(0) = 2, y' (0) = –1.

A:

Q: The solution to the IVP, is x₁ = 13x1 x2, x1(0) = x2 = 34x1+7x2, x2(0) = -1 x1(t) = e10t (cos(5t) +3…

A:

Q: -4 Find the general solution to the second order system x' " х+ -5 cos(2t). 2

A:

Q: x - ; )* ( sin 2t X' |X + 4 2 \2 cos 2t)

A: We have to solve the system of equations.

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.