Differential Equations Answers

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.

(px+y)^2=py^2

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?