What are separabledifferential equations and how to solve them?
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This is a tutorial on solving separable differential equations of the form
\[ \dfrac{dy}{dx} = \dfrac{f(x)}{g(y)} \]
Examples with detailed solutions are presented and a set of exercises is presented after the tutorials. Depending on \( f(x) \) and
\( g(y) \), these equations may be solved analytically.

Examples with Solutions

Example 1:

Solve and find a general solution to the differential equation.
\[ \dfrac{dy}{dx} = 3e^{y}x^{2} \]
Solution to Example 1:
We first rewrite the given equations in differential form and with variables separated, the y's on one side and the x's on the other side as follows.
\[ e^{-y} dy = 3x^{2} dx \]
Integrate both side.
\[ \int e^{-y} dy = \int 3x^{2} dx \]
which gives
\[ -e^{-y} + C_{1} = x^{3} + C_{2} \], \( C_{1} \) and \( C_{2} \) are constant of integration.
We now solve the above equation for y
\[ -e^{-y} = x^{3} + C_{2} - C_{1} \]
\[ e^{-y} = - x^{3} - C_{2} + C_{1} \]
\[ - y = \ln(- x^{3} - C_{2} + C_{1}) \]
\[ y = -\ln(-x^{3} - C) \], where \( C = C_{2} - C_{1} \).
As practice, verify that the solution \( y = -\ln(-x^{3} - C) \) obtained satisfy the differential equation given above.

Example 2:

Solve and find a general solution to the differential equation.
\[ \dfrac{dy}{dx} = \dfrac{\sin x}{y\cos y} \]
Solution to Example 2: Separate variables and write in differential form.
\[ y \; \cos y \;dy = \sin x \;dx \]
Integrate both sides
\[ \int y\cos y dy = \int \sin x dx \]
The left side may be integrated by parts
\[ y\; \sin y - \int \sin y \; dy = -\cos x \]
\[ y\sin y + \cos y + C_{1} = -\cos x + C_{2} \], \( C_{1} \) and \( C_{2} \) are constants of integration.
In this case there is no simple formula for \( y \) as a function of \( x \).
\[ y = \dfrac{-\cos x - \cos y + C}{\sin y} \], where \( C = C_{2} - C_{1} \)

Exercises:

Solve the following separable differential equations.
a) \( \dfrac{dy}{dx} = -9x^{2}y^{2} \)
b) \( \dfrac{dy}{dx} = -2xe^{y} \)

Solutions to the above exercises
a) \( y = \dfrac{1}{3x^{3} + C} \)
b) \( y = -\ln(x^{2} + C) \)