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With the use of reduction of order for differential equations, reduce the following to first order and solve. (i) π¦ β²β² + π π¦π¦ β²3 = 0, (ii) π₯π¦ β²β² + 2π¦ β² + π₯π¦ = 0, π¦1 = sin π₯ π₯ , (iii) (1 β π₯ 2 )π¦ β²β² β 2π₯π¦ β² + 2π¦ = 0, π¦1 = π₯, (iv) 4π₯ 2π¦ β²β² β 3π¦ = 0, π¦(1) = 3, π¦ β² (1) = 2.5.Β
Consider that an object weighing 50 lb is dropped from a height of 1000ft with zero initial velocity. Assume that the air resistance is proportional to the velocity of the body. If the limiting velocity is known to be 200ft/sec, find the time it would take for an object to reach the ground.
\left(x^3y^3+1\right)dx+x^4y^2dy=0
(px+y)^2=py^2
(D^2-5DD'+6D'^2)^2 Z=ysinx+e^2x
Solve (1+2xy)ydx+(1-2xy)xdy=0 by using inspection method
A string is stretched and fastened to two points x = 0 and x = l apart. Motion is started by
displacing the string into the form y = k(lx β x2
) from which it is released at time t = 0. Find
the displacement of any point on the string at a distance of x from one end at time t.
Hint: From this problem, we have the following boundary conditions:
y(0,t) = 0 for all t > 0
y(l,t) = 0 for all t > 0
βy
βt
(x, 0) = 0 (initial velocity is zero)
y(x, 0) = k(lx β x2
)
Given an RC series circuit that has an emf source of 50 volts, a resistance of 20k ohms, a capacitance of 6 microfarad and the initial charge of the capacitor is 1 microcoulomb. What is the charge in the capacitor at the end of 0.01 second? What is the current in the circuit at the end of 0.05 seconds?
Let u(x,y) be the harmonic function in D = {(x,y)|x2 + y2 < 36} which satisfies the Dirichlet
boundary condition
u(x,y) = x , x<0
u(x,y) = 0 , otherwise
Prove that u(x,y) < min(x,0) in D.
Evaluate u(0,0) using the mean value principle.
Using Poissonβs formula evaluate u(0,y) for 0 β€y < 6.
Using the method of separation of variables, find the solution u(x,y) in D
dx/dt=x-4y
dy/dt=x+y
#339914 on Dec 2023