Flashcards in 11.1.1 An Introduction to Differential Equations Deck (12):
An Introduction to Differential Equations
• A differential equation is an equation containing derivatives.
• Not all differential equations can be solved. Those that can be solved have infinitely many solutions.
- 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.
Which of the following is a solution of the non-separable differential equation dy/dx=(y−1)(x+y)?
y = 1
Which of the following is a solution of d2y/dx2=−6y?
Which of the following is not a solution of d2y/dx2=3y?
Which of the following is not a solution of d2y/dx2=6x?
y = x^ 3 + x ^2
Which of the following is a solution of dy/dx=x^5?
What is the order of the differential equation d2ydx2+6x3dydx+3x=y4?
Which of the following is a solution of dy/dx=1/1+x^2?
y = arctan x
Suppose you are given that f′(x)=3/2x^5−x^2+x^3.Find f(x), given that f(0)=2.
Which of the following is the general solution of dy/dx=1/x?