How do you solve separable firstorder differential equations?
A separable equation typically looks like:
By multiplying by
By integrating both sides,
which gives us the solution expressed implicitly:
where
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To solve separable firstorder differential equations:

Separate the variables by expressing the equation in the form ( \frac{{dy}}{{dx}} = g(x) \cdot h(y) ), where ( g(x) ) is a function of ( x ) only, and ( h(y) ) is a function of ( y ) only.

Integrate both sides of the equation with respect to their respective variables. This yields: [ \int \frac{{dy}}{{h(y)}} = \int g(x) , dx ]

Solve each integral independently. The integration on the left side may result in a function of ( y ), while the integration on the right side may result in a function of ( x ).

If possible, solve for ( y ) explicitly. In some cases, you may have to solve for ( y ) implicitly.

Include the constant of integration ( C ) when integrating.

If initial conditions are given, use them to determine the value of the constant of integration ( C ). If not, leave the solution in implicit form with the constant ( C ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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