Math 2414 Activity 10(Due by August 1)
Solve the following differential equations:
1. 2. 3.
4. 5. 6.
Find particular solutions for the following:
7. 8. 9.
10. Find all functions f so that all of its tangent lines pass through the origin.
{Hint: It must be that for all , or that .}
11. Find all functions such that is continuous and
for all real x.
{Hint: Differentiate both sides and use .}
12. A student forgot the Product Rule for differentiation and made the mistake of thinking that . However, he was lucky and got the correct answer. The function was , the domain of his problem was , and What was the function g?
13. Find all functions that satisfy the equation . This means that , so differentiate both sides and solve for .
14. Consider a curve that passes through the origin, with increasing, differentiable on , and , so that when horizontal and vertical lines are each drawn from the point with to the coordinate axes, the area under the curve is twice the area above the curve.
Find all the functions that satisfy these conditions.
Start by showing that and .
Or if you prefer,.
15. A snowball melts at a rate proportional to its surface area. If the snowball originally has a radius of 10 cm, and after 20 minutes, its radius is 8 cm, find
a) a differential equation for the radius, r.
{Hint: The first sentence translates into , but , , where V is the volume of the snowball and S is its surface area. We have that , so solve for .}
b) Solve the previous DE subject to and .
c) What’s the radius of the snowball after one hour?
d) How long will it take for the snowball to completely melt?
16. Find all continuous functions f so that .
{Hint: Differentiate the integral equation and then use the integral equation to determine an initial condition.}
17. a) A dog spots a rabbit running in a straight line. Fortunately for the rabbit, the distance between the dog and the rabbit remains constant. Assuming that the dog always moves towards the rabbit, and the distance between them is always 10 feet, find an equation to describe the path of the dog.
{Hint:
This leads to the differential equation and initial condition
for the dog’s path.}
b) Now assume that the dog starts at the point , the rabbit starts at and has constant speed, and the dog moves toward the rabbit at twice the speed of the rabbit. Find an equation to describe the dog’s path.
{Hint:
and
Solving for t in each and equating leads to . Differentiating leads to or }
c) How far does the rabbit run in part b) before the dog catches it?
18. Consider the function . The improper integral is convergent for . In fact F is differentiable for , with . (Notice that .) Here’s why:
So .
Since , the Sandwich Theorem implies that
.
Using integration by parts, you can show that . So F satisfies the differential equation . Using methods from Calculus III, you can show that . Solve subject to to find a much simpler formula for F.
19. The differential equation is not separable, but if you make the substitution , you get , which is separable in the variables x and u. Solve the new separable equation and use it to solve the original equation.
20. The differential equation is not separable, but if you solve for , you get . If you make the substitution , you get , which is separable in the variables x and u. Solve the new separable equation and use it to solve the original equation.
21. Consider . Since and
, the integral is proper.
a) Show that . {Hint: , and problem #18}
b)Use the fact that to find a simple formula for .
c) Use the previous results to find the exact value of .
d)Use the previous results to find the exact value of .
22. The function is a solution to the initial value problem . Explain why it is the only solution.
{Hint: Suppose that , and consider the function . Compute , and draw a conclusion.}
23. For what values of is a solution of the differential equation ?
24. Find a differentiable function f, which isn’t identically zero, that satisfies the integral equation .