Worksheet 11.1.Intro to diff eq PDF

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Worksheet 11.1.Intro to diff eq...

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106L Worksheets: Introduction to Differential Equations

Introduction to Differential Equations 1. (a) Is the point x = 1, y = 3 (i.e. the point (1, 3)) a solution to the equation 2y + x = y + 4? Why?

(b) Is the point (2, 3) a solution to the equation y2 + x = y + 8? Why?

(c) Is the point (1, 1) a solution to the equation x2 + y2 = 1? Why?

(d) What does it mean to say that a point (a, b) is a solution to an equation?

2. (a) Is the function y = x2 a solution to the equation

dy dx

(b) Is the function y = − 1x a solution to the equation

x2 2

a solution to the equation If y = If y = If y =

x2

dy dx

= y2 ?

then dy = dx then y2 = is dy = y2 ? dx

If y = − 1x , If y = − 1x , If y = − 1x , (c) Is the function y =

= 2x? Why?

dy dx

= y + x?

then dy = dx then x + y = is dy = y + x? dx

,

2 x2 , 2 x2 , 2

3. (a) Is the function y = e3x a solution to the equation If y = e3x , If y = e3x ,

dy dx

then dy = dx then 3y =

(b) Is the function y = e3x + 1 a solution to the equation If y = e3x + 1, If y = e3x + 1,

1

= 3y ?

then dy = dx then 3y =

dy dx

= 3y ?

106L Worksheets: Introduction to Differential Equations (c) Is the function y = 2e3x a solution to the equation

dy dx

= 3y ?

then dy = dx then 3y =

If y = 2e3x , If y = 2e3x ,

dy dy Equations of the form dx = y2 , dy = 2x or dx = y + x are called differential dx dy equations. In fact, any equation of the form dt = g(y, t) is a differential equation. y = f (t) is a solution if when f (t) is substituted for y in the expression g(y, t), the result is dy . In other words, like any other equation, when you substitute your dt answer into both sides of the equation, you get a true statement.

4. Find a function that satisfies the equation

dy dx

= 3 x.

dy dx

= ky. (Hint: see Question 3, part (a))

Can you name another such function?

5. Find a function that satisfies the equation

Can you name another such function?

6. Which of the following is a solution to the differential equation (a) f (t) = e9t

(b) f (t) = 5e9t

(c) f (t) = e3t

(b) f (t) = a2 t2 + 3

2

= 9y ?

(d) f (t) = e9t + 5

7. Which of the following is a solution to the differential equation (a) f (t) = a2 t2

dy dt

(c) f (t) = eat

dy dt

= at?

(d) f (t) = (at)2

106L Worksheets: Introduction to Differential Equations 8. Which of the following is a solution to the differential equation

dy dt

= 2(y − 1)?

(a) 1 − e2t

(b) 1 + e2t

(c) e2t − 2t

(d) 2 + e2t

In general, differential equations have an infinite number of solutions, but if we are given an initial starting point, there will only be one solution.

A system consisting of a differential equation and an initial condition is called an initial value problem

9. Which of the following is the solution to the initial value problem dy = 9y with y = 3 at t = 0? dt (a) f (t) = e9t

(b) f (t) = 3e9t

(c) f (t) = 2e9t

3

(d) f (t) = 4e9t

(e) f (t) = 2e9t + 1

106L Worksheets: Introduction to Differential Equations 10. Which of the following is the solution to the initial value problem dy = 5t with y = 4.5 at t = 1? dt (a) f (t) = 52 t2

(b) f (t) = 52 t2 + 2

11. Solve the initial value problem

dy dt

(c) f (t) = 5t2 +

1 2

(d) f (t) = 52 t2 + 0.5

= 6y with

(a) y = 2 at t = 0

(b) y = 2 at t = 1

12. Find a function of the form y = xn that is a solution to the differential equation

4

1 x dy 2 dx

= y....