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

• A separable differential equation is one where the variables can be separated into two functions, one with only the independent variable and one with only the dependent variable
• What is the first step in identifying and checking if a differential equation is separable?

  Rearrange the equation
• If the left side of a rearranged differential equation contains only $y$ terms and the right side contains only $x$ terms, the equation is separable.


  True
• To separate variables in $\frac{dy}{dx} =$$\frac{x}{y}$, you first rearrange it to y\, dy = x\, dx</latex>, then group $y$ terms with \dy and $x$ terms with \dx before integrating.
• Once a differential equation is identified as separable, the method of separation of variables can be applied


  True
• The equation `dy/dx = x/y` is rearranged to isolate dy and dx as y dy = x dx
• After separating variables, the next step is to integrate both sides of the equation


  True
• The resulting integrals after integrating both sides of a separable differential equation give the general solution


  True
• A separable differential equation can always be written in the form $\frac{dy}{dx} =$$f(x)g(y)$


  True
• What is the primary goal of rearranging the equation in the first step of checking separability?

  Isolate dependent variable
• What is the purpose of separating variables in a differential equation?

  Prepare for integration
• A differential equation is separable if it can be written as f(y) dy = g(x) dx
• Match the action with its example in separating variables:

  Rearrange the equation ↔️ dy/dx = x/y becomes y dy = x dx

  Group terms ↔️ ∫ y dy = ∫ x dx

  Integrate both sides ↔️ (1/2)y^2 = (1/2)x^2 + C
• The general solution to a separable differential equation expresses the dependent variable in terms of the independent variable and the constant of integration
• Separating variables allows you to solve for the dependent variable in terms of the independent variable and the constant of integration.

  True
• After separating variables in a separable differential equation, the next step is to integrate both sides
• Match the steps to find the general solution of a separable differential equation with their descriptions:

  Identify and Check ↔️ Determine if the equation is separable by isolating the dependent variable.

  Separate Variables ↔️ Rearrange the equation to group terms with dy on one side and dx on the other.

  Integrate Both Sides ↔️ Apply the power rule for integration to get the general solution.

  Solve for Dependent Variable ↔️ Express y in terms of x and the constant C.
• Steps to solve a separable differential equation

  1️⃣ Identify if the equation is separable

  2️⃣ Separate the variables

  3️⃣ Group terms with dy and dx

  4️⃣ Integrate both sides

  5️⃣ Solve for the dependent variable
• Steps to separate variables in a separable differential equation

  1️⃣ Rearrange: Isolate the terms with the dependent variable (dy) on one side and the independent variable (dx) on the other.

  2️⃣ Group: Put the y terms with dy on one side and the x terms with dx on the other.

  3️⃣ Integrate: Integrate both sides of the equation.
• Integrating both sides of a separable differential equation produces the general solution, which contains a constant of integration.


  True
• The general solution of a separable differential equation is in the form F(y) = G(x) + C
• A separable differential equation is one where the variables can be separated into two functions, one with only the independent variable and one with only the dependent variable
• Steps to separate variables in a differential equation

  1️⃣ Rearrange the equation to isolate terms involving \dy and \dx

  2️⃣ Group $y$ terms with \dy on one side and $x$ terms with \dx on the other

  3️⃣ Integrate both sides of the equation
• What is a separable differential equation?

  Variables can be separated
• What is the first step to separate variables in a separable differential equation?

  Rearrange the equation
• What does separating variables set up the differential equation for?

  Integration
• To integrate $\int y^{n} dy$, the power rule for integration states that the result is (1/(n+1))y^{n+1} + C
• What is the final step in solving a separable differential equation?

  Solve for the dependent variable
• A separable differential equation is one where the variables can be separated into two functions, one with only the dependent variable
• Steps to solve for the dependent variable in a separable differential equation

  1️⃣ Separate the variables: Isolate the terms with dy on one side and dx on the other.

  2️⃣ Integrate both sides: Apply the power rule for integration to get the general solution.

  3️⃣ Solve for the dependent variable: Rearrange the general solution to express y in terms of x and C.
• The general solution of a separable differential equation expresses the dependent variable in terms of the independent variable and the constant of integration.


  True