How do you solve separable first-order differential equations?

Question

William Abdalla · Accepted Answer

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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William Abdalla · Answer

undefined To solve separable first-order differential equations:

1. 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.

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

3. 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 $.

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

5. Include the constant of integration $ C $ when integrating.

6. 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 $.