StudyXY
RegisterLog in
Back to AI Flashcard MakerMathematics /AP Calculus AB: 11.1.2 Solving Separable Differential Equations

AP Calculus AB: 11.1.2 Solving Separable Differential Equations

Mathematics13 CardsCreated 3 months ago

This content explains how to solve separable differential equations by separating variables and integrating both sides. It highlights the use of initial conditions to find particular solutions, discusses the role of constants of integration, and shows how varying constants produces families of solution curves.

PrintEmbedImportReport

Solving Separable Differential Equations

  • To solve a separable differential equation, collect the x-terms with the dx differential and the y-terms with the dy differential.

  • Initial conditions allow you to find a particular solution to a differential equation.

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/13

Key Terms

Term
Definition

Solving Separable Differential Equations

  • To solve a separable differential equation, collect the x-terms with the dx differential and the y-terms with the dy differential.

    • ...

note

  • Check whether this differential equation is separable.
    Multiply both sides of the equation by dx.

  • If you can gather all t...

Which of the following is a solution of dy/dx=y^2?

y=−1/x+1

Which of the following is the general solution of dy/dx=xe^x+y ?

–e^−y=xex−ex+C

Which of the following is the general solution of dydx=x/y^2+1 ?

x^2/2=y^3/3+y+C1

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

y^3/3=x^3/3+x^2/2+C

Log in to view all terms

Related Flashcard Decks

Mathematics

Essential SAT Math Formulas

This deck covers key formulas and concepts essential for the SAT Math section, including slope, distance, and vertex forms.

10 cards
View Deck
Mathematics

Brain and Behavior Chapter 4: Neural Conduction & Synaptic Transmission Part 4

This deck covers key concepts from Chapter 4 of Brain and Behavior, focusing on neural conduction and synaptic transmission, including neurotransmitter deactivation, glial cell functions, and neurotransmitter classifications.

25 cards
View Deck
Mathematics

Behavioral Neuroscience Chapter 3 Test

This deck covers key concepts from Chapter 3 of Behavioral Neuroscience, focusing on the structure and function of the nervous system, including sensory neurons, brain regions, and neural techniques.

20 cards
View Deck
Mathematics

Algebra 1 Unit 1 Mastery Test

These properties of equality help maintain balance in an equation when performing the same operation on both sides. Inverse operations, like using addition to undo subtraction, are key tools in solving equations.

29 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 2

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

53 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 1

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

55 cards
View Deck

Study Tips

  • Press F to enter focus mode for distraction-free studying
  • Review cards regularly to improve retention
  • Try to recall the answer before flipping the card
  • Share this deck with friends to study together
AP Calculus AB: 11.1.2 Solving Separable Differential Equations
TermDefinition

Solving Separable Differential Equations

  • To solve a separable differential equation, collect the x-terms with the dx differential and the y-terms with the dy differential.

  • Initial conditions allow you to find a particular solution to a differential equation.

note

  • Check whether this differential equation is separable.
    Multiply both sides of the equation by dx.

  • If you can gather all the x-terms with dx and all the y-terms with dy, then the differential equation is separable. You can then integrate each side of the equation.

  • Each integral will have its own constant of integration, but you can combine them into a single constant C.

  • This general solution can even be solved for y.

  • Varying the value of the constant produces multiple solutions. You can graph them to form the solution curves.

  • If you know the value of y corresponding to a given value of x, then you use can use this initial condition to determine a particular solution.

  • When you plug in the values of x and y, the only unknown is C.

  • Solve for C and plug it into the general solution to produce a particular solution.

  • Multiply both sides of this differential equation by dx to
    determine whether it is separable.

  • It is separable, so integrate both sides.

  • The two constants of integration combine to form one constant in the general solution. Varying the value of C produces a family of solution curves.

  • General solutions do not always have to be solved for y.

Which of the following is a solution of dy/dx=y^2?

y=−1/x+1

Which of the following is the general solution of dy/dx=xe^x+y ?

–e^−y=xex−ex+C

Which of the following is the general solution of dydx=x/y^2+1 ?

x^2/2=y^3/3+y+C1

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

y^3/3=x^3/3+x^2/2+C

Which of the following is the general solution of y2+y2x2+xdydx=1?

2x33+x22=y33+y22+C

Which of the following is the general solution of dydx=xy2y4+1 ?

x22=y33−1y+C

Which of the following is the general solution of dydx=xy?

y 2 = x 2 + C

Which of the following is the general solution of dydx=y2(x3+x2+1)?

−1y=x44+x33+x+C

Which of the following is the general solution of dydx=2xy2?

y=−1x2+C

Which of the following is the general solution of dydx=sinxy+1?

y22+y=C−cosx

Which of the following is the general solution of dydx=xy2−y2+x−1?

arctany=x22−x+C