How do you solve separable differential equations?

Answer 1

A separable equation typically looks like:
#{dy}/{dx}={g(x)}/{f(y)}#.

By multiplying by #dx# and by #g(y)# to separate #x#'s and #y#'s,
#Rightarrow f(y)dy=g(x)dx#

By integrating both sides,
#Rightarrow int f(y)dy=int g(x)dx#,
which gives us the solution expressed implicitly:

#Rightarrow F(y)=G(x)+C#,
where #F# and #G# are antiderivatives of #f# and #g#, respectively.

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Answer 2

To solve a separable differential equation:

  1. Separate the variables: Write the equation so that all terms involving the dependent variable (usually y) are on one side, and all terms involving the independent variable (usually x) are on the other side.

  2. Integrate both sides with respect to their respective variables.

  3. Solve for the constant of integration, if necessary, using any initial conditions given.

  4. Express the solution as a function of the dependent variable (y) in terms of the independent variable (x).

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Answer 3

To solve separable differential equations, follow these general steps:

  1. Separate the variables: Write the differential equation in a form where all terms involving the dependent variable and its derivative are on one side, and all terms involving the independent variable are on the other side.

  2. Integrate both sides: Integrate each side of the equation with respect to its corresponding variable.

  3. Solve for the constant: If necessary, use initial conditions or boundary conditions to determine the value of the constant of integration.

  4. Write the general solution: Combine the results of the integrations, including the constant of integration, to form the general solution.

  5. If given initial conditions: Use the initial conditions to find the particular solution by substituting the values into the general solution and solving for the constant.

  6. Check for extraneous solutions: Ensure that any solutions obtained satisfy the domain of the original differential equation.

These steps can be applied to a wide range of separable differential equations encountered in various fields of mathematics and science.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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