Differential Equations Answers

The population of a certain state is known to grow at a rate proportional to the number of people presently living in the state. If after 10 years the population has trebled and if after 20 years the population is 150,000, find the number of people initially living in the state

Under certain conditions, cane sugar is converted into dextrose at a rate, which is proportional to the amount unconverted at any time. If out of 60 grams of sugar at t = 0 6 grams are converted during the first 4 minutes, find the amount converted in 2 hours.

1. Solve the following Bernoulli's Differential Equations. Show your solutions.

a. dy/dx + (1/3) y = e^x y²

b. x (dy/dx) + y = xy³

c. dy/dx + (2/x) y = -x² cos x y²

d. x²y-x³ (dy/dx) = y² cos x

(2xy+3y2)dx-(2xy+X2)dy=0

A condenser of capacity C=5*10^5 farad is charged through a resistance R=200ohms by steady voltage E=2000volts.Calculate the current at the instant of closing the switch

Use the method of variation of parameters to solve the differential equation:

y"-2y'+y=xe^xtan-1x

Solve the following differential equation by using the method of undetermined

coefficients:

𝑦"+4𝑦=3𝑥+𝑐𝑜𝑠(2𝑥)

[D] y''-25y=0; y1=e^5x the indicated function y1(x) is a solution of the given differential eqution.Use reduction of order or formula as instructed, to find a second solution y2(x).

The lines of electric force of two opposite charges of the same strength at (-1,0) and (1,0) are the circles through (-1,0) and (1,0). Show that these circles are given by x^2 + (y - c)^2 = 1 + c^2. Show that the equipotential lines (which are orthogonal trajectories of those circles) are the circles given by (x + c*)^2 + y^2 = c*^2 - 1.

Let the electric equipotential lines (curves of constant potential) between two concentric cylinders with the z-axis in space be given by u(x,y) = x^2:+:y^2 = c (these are circular cylinders in xyz-space). Using the method in the text, find their orthogonal trajectories (the curves of electric force).