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AP Calculus AB: 11.1.1 An Introduction to Differential Equations

Mathematics12 CardsCreated 3 months ago

This content introduces differential equations, focusing on their definitions, general and particular solutions, and the concept of order. It explains separable equations, initial conditions, and solution methods, with examples of solving both first- and second-order differential equations.

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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.

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Key Terms

Term
Definition

An Introduction to Differential Equations

  • A differential equation is an equation containing derivatives.

  • Not all differential equations can be solved. Those that can ...

note

  • In this equation, you want to solve for the function y whose derivative is e x . This is a differential equation.

  • Since the ...

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?

y=sin(√6x)

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

y=sin√3x

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

y = x^ 3 + x ^2

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AP Calculus AB: 11.1.1 An Introduction to Differential Equations
TermDefinition

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.

note

  • 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?

y=sin(√6x)

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

y=sin√3x

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?

y=x^6/6−1

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

second-order

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.

f(x)=1/4x^6−1/4x^2+1/4x^4+2

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

y=lnx+C

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

y = e ^x