7.6 Finding General Solutions Using Separation of Variables

• What is a separable differential equation?

  Variables can be separated
• If an equation can be written as dy/dx = f(x)g(y), it is a separable differential equation.

  True
• In a non-separable differential equation, the variables x and y cannot be separated.
• Give an example of a non-separable differential equation.

  dy/dx = x^2 + y^2
• What is the first step in separating variables in a differential equation?

  Rearrange the equation
• Non-separable differential equations cannot be solved by separating the variables.


  True
• When integrating dy/y, the result is ln|y| + C1.
• What operation is performed on both sides of the equation to eliminate the natural logarithm?

  Exponentiation
• What is the final step after separating variables and integrating both sides of a separable differential equation?

  Solve for y
• Exponentiating both sides eliminates the natural logarithm
• A separable differential equation is one where variables x and y can be separated into two separate functions


  True
• What is the key difference between separable and non-separable differential equations?

  Variables can be separated
• What is the simplified form of dy/dx = xy after separating the variables?

  dy/y = x dx
• After integrating both sides, add a constant of integration
• The final step in solving a separable differential equation is to solve for the dependent variable y
• To identify a separable differential equation, rearrange it into the form dy/dx = f(x)g(y)
• What is the key difference between separable and non-separable differential equations?

  Variables can be separated
• Separable differential equations can be solved by integrating the two separate functions.


  True
• Steps to identify a separable differential equation

  1️⃣ Rearrange the equation to dy/dx = f(x)g(y)

  2️⃣ Check if f(x) and g(y) are separate functions

  3️⃣ If yes, it is separable
• When separating variables, the goal is to write the equation in the form dy/dx = f(x)g(y)
• What is added to one side of the equation after integrating both sides in the separation of variables method?

  Constant of integration
• The constant of integration can be combined into a single constant after simplification.


  True
• After exponentiating both sides, simplify the expression to combine the constant of integration into a single constant.
• Steps to solve for the dependent variable y

  1️⃣ Isolate the natural logarithm term

  2️⃣ Exponentiate both sides

  3️⃣ Combine the constant of integration
• What does the constant of integration K represent in the general solution of a separable differential equation?

  Initial condition
• A separable differential equation can be written in the form dy/dx = f(x)g(y)
• In a separable differential equation, variables x and y cannot be separated into two separate functions

  False
• Steps to integrate both sides of a separable differential equation

  1️⃣ Integrate both sides with respect to their variables

  2️⃣ Add a constant of integration

  3️⃣ Simplify the expression
• What is the first step to solve for y after separating variables and integrating both sides?

  Isolate the natural logarithm
• What is the first step in solving for y in a separable differential equation?

  Isolate ln|y|
• The constant of integration C can be combined into a single constant K
• Finding the general solution to a separable differential equation completes the process of solving it.


  True
• Steps to separate the variables in a separable differential equation

  1️⃣ Rearrange the equation to the form dy/dx = f(x)g(y)

  2️⃣ Express the equation as separate functions of x and y

  3️⃣ Verify that the equation is separable
• After integrating both sides of a separable differential equation, add a constant of integration C
• The general solution to a separable differential equation is often in the form $y =$$Ke^{f(x)}$


  True
• To eliminate the natural logarithm in the equation $\ln|y| =$$\frac{x^{2}}{2} +$$C$, you exponentiate both sides.


  True
• What does the constant of integration K represent in the general solution of a separable differential equation?

  Initial condition of y
• What is the next step after separating the variables in a separable differential equation?

  Integrate both sides
• Steps to find the general solution to a separable differential equation

  1️⃣ Separate the variables

  2️⃣ Integrate both sides

  3️⃣ Add the constant of integration

  4️⃣ Simplify the expression

  5️⃣ Solve for the dependent variable y