Applying the method of separation of variables to solve differential equations:

Cards (43)

  • Differential equations describe relationships between a function and its rate of change
  • Steps in the method of separation of variables
    1️⃣ Separate variables
    2️⃣ Integrate both sides
    3️⃣ Solve for y
  • The general solution to a differential equation includes arbitrary constants.

    True
  • In separation of variables, terms involving y and dy are moved to one side of the equation.
    True
  • Separable differential equations can be written in the form dy/dx = f(x)g(y).

    True
  • In a separable differential equation, g(y)g(y) contains only the variable yy.

    True
  • Steps to solve a separable differential equation
    1️⃣ Separate the variables
    2️⃣ Integrate both sides
    3️⃣ Solve for yy
  • Match the type of solution with its definition:
    General Solution ↔️ Includes arbitrary constants
    Particular Solution ↔️ Obtained from initial conditions
  • Steps to identify a separable differential equation
    1️⃣ Check if it can be written as \frac{dy}{dx} = f(x)g(y)</latex>
    2️⃣ Verify that f(x)f(x) depends only on xx
    3️⃣ Verify that g(y)g(y) depends only on yy
  • A separable differential equation can be rewritten in the form \frac{dy}{dx}
  • An example of a separable differential equation is \frac{dy}{dx} = xy
  • What is the next step after separating the variables in a separable differential equation?
    Integrate both sides
  • What does the arbitrary constant C in the general solution represent?
    A family of functions
  • The particular solution is found by applying initial conditions to the general solution.

    True
  • How do you verify that y=y =2ex22 2e^{\frac{x^{2}}{2}} is a solution to dydx=\frac{dy}{dx} =xy xy?

    Substitute and simplify
  • A particular solution to a differential equation includes arbitrary constants.
    False
  • A particular solution is derived from the general solution by applying initial conditions.
    True
  • Match the solution type with its example:
    General Solution ↔️ y = x^2/2 + Cx
    Particular Solution ↔️ y = x^2/2 + 2x (given y(0) = 0)
  • The general solution contains specific constant values derived from initial conditions.
    False
  • Match the characteristic with the type of differential equation:
    Separable ↔️ Variables can be isolated
    Non-Separable ↔️ Variables cannot be isolated
  • To separate variables, move yy terms to one side and xx terms to the other, then multiply or divide to isolate dy and dxdx.
  • After integrating dyy=\int \frac{dy}{y} =xdx \int x \, dx, the result is lny=\ln|y| =x22+ \frac{x^{2}}{2} +C C, where CC is an arbitrary constant.
  • The method of separation of variables involves separating variables so that all yy terms are on one side and xx terms are on the other.
  • An example of a non-separable differential equation is dydx=\frac{dy}{dx} =xy+ xy +y2 y^{2}, as the variables cannot be isolated.
  • In a separable differential equation, f(x) contains only the independent variable x.

    True
  • Steps to separate the variables in a separable differential equation:
    1️⃣ Move the y-terms to one side
    2️⃣ Move the x-terms to the other side
    3️⃣ Isolate dy and dx
  • The general solution for dyy=\int \frac{dy}{y} =xdx \int x \, dx is \ln|y| = \frac{x^{2}}{2} + C</latex>.

    True
  • Steps to solve for the dependent variable y in a general solution:
    1️⃣ Simplify the expression
    2️⃣ Solve for y by exponentiating
  • If the initial condition is y(0) = 2 and the general solution is y=y =Cex22 Ce^{\frac{x^{2}}{2}}, the particular solution is y = 2e^{\frac{x^{2}}{2}}
  • The separation of variables technique simplifies integration by isolating variables on opposite sides of the equation
  • Differential equations describe the relationship between a function and its rate of change
  • The separation of variables technique is used to solve certain types of differential equations by isolating variables on opposite sides of the equation
  • A separable differential equation is one where variables can be separated into two sides of the equation
  • A separable differential equation can be rewritten in the form dydx=\frac{dy}{dx} =f(x)g(y) f(x)g(y), where f(x)f(x) contains only the variable x
  • The separated variables in the equation dydx=\frac{dy}{dx} =xy xy are dyy=\frac{dy}{y} =xdx x \, dx.

    True
  • The general solution of the differential equation dydx=\frac{dy}{dx} =xy xy is y=y =Cex22 Ce^{\frac{x^{2}}{2}}.

    True
  • The integration step in separation of variables results in the general solution.
    True
  • What are separable differential equations characterized by?
    Variables can be separated
  • What is a key difference between separable and non-separable differential equations?
    Variables can be isolated
  • To separate the variables in dydx=\frac{dy}{dx} =xy xy, we rewrite it as \frac{dy}{y} = x \, dx