11.1.1 An Introduction to Differential Equations

Question 1

An Introduction to Differential Equations

Answer 1

• A differential equation is an equation containing derivatives.
• Not all differential equations can be solved. Those that can be solved have infinitely many solutions.

Question 2

note

Answer 2

• In this equation, you want to solve for the function y whose derivative is e x . This is a differential equation.
• Since the derivative of e x is e x , y = e x is one solution of the differential equation. By adding a constant C you can form a family of solutions also called the general solution.
• Sometimes a differential equation will come with extra
  information, such as an initial condition.
• The initial condition allows you to determine the value of the constant C.
• In this case, y = 2 when x = 0. Plugging those values into the general solution produces a value of one for C.
• The general solution determines a family of curves known as the solution curves. Here are the graphs of some of these curves.
• For an equation with an initial condition, there is one solution, and it corresponds to one curve.
• The order of a differential equation is determined by the highest derivative involved.
• Typically, as the order of a differential equation gets higher, it becomes more complicated to solve.
• Here is some “fantasy math.”
• Break up the derivative symbol into two differentials and multiply to get dx on the right side.
• Now divide both sides by e –y . This produces an equation with only y-expressions on the left and only x-expressions on the right. Since the variables can be collected with their differentials, the differential equation is separable.

Question 3

Which of the following is a solution of the non-separable differential equation dy/dx=(y−1)(x+y)?

Answer 3

y = 1

Question 4

Which of the following is a solution of d2y/dx2=−6y?

Answer 4

y=sin(√6x)

Question 5

Which of the following is not a solution of d2y/dx2=3y?

Answer 5

y=sin√3x

Question 6

Which of the following is not a solution of d2y/dx2=6x?

Answer 6

y = x^ 3 + x ^2

Question 7

Which of the following is a solution of dy/dx=x^5?

Answer 7

y=x^6/6−1

Question 8

What is the order of the differential equation d2ydx2+6x3dydx+3x=y4?

Answer 8

second-order

Question 9

Which of the following is a solution of dy/dx=1/1+x^2?

Answer 9

y = arctan x

Question 10

Suppose you are given that f′(x)=3/2x^5−x^2+x^3.Find f(x), given that f(0)=2.

Answer 10

f(x)=1/4x^6−1/4x^2+1/4x^4+2

Question 11

Which of the following is the general solution of dy/dx=1/x?

Answer 11

y=lnx+C

Question 12

Which of the following is not a solution of dy/dx=−e^x?

Answer 12

y = e ^x