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

• 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, $g(y)$ contains only the variable $y$.


  True
• Steps to solve a separable differential equation

  1️⃣ Separate the variables

  2️⃣ Integrate both sides

  3️⃣ Solve for $y$
• 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 $f(x)$ depends only on $x$

  3️⃣ Verify that $g(y)$ depends only on $y$
• 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 =$$2e^{\frac{x^{2}}{2}}$ is a solution to $\frac{dy}{dx} =$$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 $y$ terms to one side and $x$ terms to the other, then multiply or divide to isolate dy and $dx$.
• After integrating $\int \frac{dy}{y} =$$\int x \, dx$, the result is $\ln|y| =$$\frac{x^{2}}{2} +$$C$, where $C$ is an arbitrary constant.
• The method of separation of variables involves separating variables so that all $y$ terms are on one side and $x$ terms are on the other.
• An example of a non-separable differential equation is $\frac{dy}{dx} =$$xy +$$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 $\int \frac{dy}{y} =$$\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 =$$Ce^{\frac{x^{2}}{2}}$, the particular solution is y = 2e^{\frac{x^{2}}{2}}
• The separation of variables technique simplifies integration by isolating variables on opposite sides of the equation
• Differential equations describe the relationship between a function and its rate of change
• The separation of variables technique is used to solve certain types of differential equations by isolating variables on opposite sides of the equation
• A separable differential equation is one where variables can be separated into two sides of the equation
• A separable differential equation can be rewritten in the form $\frac{dy}{dx} =$$f(x)g(y)$, where $f(x)$ contains only the variable x
• The separated variables in the equation $\frac{dy}{dx} =$$xy$ are $\frac{dy}{y} =$$x \, dx$.


  True
• The general solution of the differential equation $\frac{dy}{dx} =$$xy$ is $y =$$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 $\frac{dy}{dx} =$$xy$, we rewrite it as \frac{dy}{y} = x \, dx