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Solve the following boundary value problems.
(i) 𝑦
′′ + 4𝑦 = 0; 𝑦(0) = 3, 𝑦(𝜋/2) = −3,
(ii) 𝑦
′′ − 25𝑦 = 0; 𝑦(−2) = 𝑦(2) = cosh 10.
(iii) 𝑦
′′ + 2𝑦
′ + 2𝑦 = 0; 𝑦(0) = 1, 𝑦(𝜋/2) = 0.
With the use of reduction of order for differential
equations, reduce the following to first order and
solve.
(i) 𝑦
′′ + 𝑒
𝑦𝑦
′3 = 0,
(ii) 𝑥𝑦
′′ + 2𝑦
′ + 𝑥𝑦 = 0, 𝑦1 =
sin 𝑥
𝑥
,
(iii) (1 − 𝑥
2
)𝑦
′′ − 2𝑥𝑦
′ + 2𝑦 = 0, 𝑦1 = 𝑥,
(iv) 4𝑥
2𝑦
′′ − 3𝑦 = 0, 𝑦(1) = 3, 𝑦
′
(1) = 2.5
With the use of reduction of order for differential equations, reduce the following to first order and solve. (i) 𝑦 ′′ + 𝑒 𝑦𝑦 ′3 = 0, (ii) 𝑥𝑦 ′′ + 2𝑦 ′ + 𝑥𝑦 = 0, 𝑦1 = sin 𝑥 𝑥 , (iii) (1 − 𝑥 2 )𝑦 ′′ − 2𝑥𝑦 ′ + 2𝑦 = 0, 𝑦1 = 𝑥, (iv) 4𝑥 2𝑦 ′′ − 3𝑦 = 0, 𝑦(1) = 3, 𝑦 ′ (1) = 2.5.
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 − 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
∂y
∂t
(x, 0) = 0 (initial velocity is zero)
y(x, 0) = k(lx − 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?
