Sample Questions: First Order Equations

1.

Show that y is an integrating factor of $$2 x y^2 - y + (3 x^2 y - 2 x + 2)\, y' = 0$$. Find the solutions F(x,y) = constant.   Solution

2.

Find the general solutions of

(a) $$x y' = (1-y^2)^{1/2}$$   Solution

(b) $$y' + \frac{2 y}{x} = \frac{\sin x}{x^2}$$  Solution

(c) $2 x \sin y + (1 + x^2 \cos y) y' = 0$  Solution

(d) $$\frac{dy}{dx} = \frac{x y + y}{xy + x}$$  Solution

3.

Initially a tank contains 1 litre of water and 2 grams of salt. A mixture containing 1 gram/litre flows in at the rate of 2 litres/min. The stirred mixture from the tank flows out at 1 litre/min. Find the concentration of salt in the tank as a function of time. What is the limiting concentration for large time (and a large tank)?   Solution

4.

Find (if possible) a solution that satisfies the initial condition y(0) = 1.

(a) $$y' = y \tan x - 2 \sin x$$   Solution

(b) $$y' = \frac{xy}{1+y^2}$$  Solution

(c) $$x^2 y + y^2 + (x^3 - 2 x y) y' = 0$$  Solution

5.

An object of mass m is dropped from rest subject to the gravitational force mg and also an air resistance force proportional to the magnitude of the velocity.

(a) Find the velocity of the object at any time t.   Solution

(b) Find the time at which the velocity reaches 80% of its limiting velocity as $t \to \infty$. Leave your answer expressed in terms of physical constants and logarithms.   Solution

6.

Solve the initial value problem $$y'' = 12 y^2$$, y(0) = 2, y'(0) = 8. (Hint: write a differential equation for v = y' as a function of y). On what interval is the solution defined?   Solution

7.

The curve shown here constitutes two solutions of one of the following three differential equations. Why two solutions rather than one? Which equation is it? State your reasons. Note: don't try to solve the equations.   Solution

(a) $$\frac{dy}{dx} = \frac{x y^2}{1 + y^2}$$

(b) $$\frac{dy}{dx} = \frac{x y}{1 - x y}$$

(c) $$\frac{dy}{dx} = \frac{x y}{x - y}$$

8.

Solve the initial value problems:

(a) $$\frac{dy}{dt} = \frac{yt}{(1+y)(1+t)},\quad y(0) = 1$$  Solution

(b) $$(x^2 - y^2)\, dx - 2 y\, dy = 0,\quad y(2) = 1$$

Hint: the equation can be made exact by multiplication by an integrating factor which depends only on x   Solution

9.

In a certain animal population, the number of births per unit time is proportional to the population, but the number of deaths per unit time is proportional to the square of the population. When the population was $10\,000$, there were $20\,000$ births per year and $10\,000$ deaths per year.

(a) What happens to the population in the long run?

(b) How long does it take for the population to go from $10\,000$to $15\,000$?

  Solution