The World of Separable Differential Equations
Separable differential equations are a type of first-order ODE where the variables can be separated on either side of the equation. By rearranging, all terms with one variable go on one side, and all terms with the other variable on the opposite side, allowing for straightforward integration and solution.
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Solving separable differential equations:
To solve a separable differential equation, the variables \( y \) and \( x \) are first separated. This is followed by integrating both sides of the equation independently. After integration, an arbitrary constant is introduced, and the equation is rearranged to derive the general solution, expressing the relationship between \( y \), \( x \), and the constant \( C \).
Here is an example:
\( \text{Problem: Solve the separable differential equation }\)
\(\frac{dy}{dx} = \frac{y^2 \sin(x)}{e^y}, \text{ where } y \neq 0.\)
\(\text{First, separate the variables } y \text{ and } x:\)
\(e^y dy = y^2 \sin(x) dx.\)
\(\text{Next, integrate both sides of the equation:}\)
\(\int e^y dy = \int y^2 \sin(x) dx.\)
\(\text{Perform the integrations:}\)
\(e^y = -y^2 \cos(x) + g(y),\)
\(\text{where } g(y) \text{ is an arbitrary function of } y.\)
\(\text{Since the left side is a function of } y \text{ only, } g(y) \text{ must be a constant:}\)
\(e^y = -y^2 \cos(x) + C.\)
\(\text{The general solution to the differential equation is:}\)
\(e^y + y^2 \cos(x) = C,\)
\(\text{where } C \text{ is an arbitrary constant.} \)
Here is another problem:
\( \text{Problem: Solve the differential equation }\)
\(\frac{dy}{dx} = \frac{\cos(x) + \sin(y)}{y(1 + x^2)}, \text{ with } y \neq 0 \text{ and } x \neq 0.\)
\(\text{Integrate both sides:}\)
\(\int y(1 + x^2) dy = \int (\cos(x) + \sin(y)) dx.\)
\(\text{Performing the integrations:}\)
\(\frac{y^2}{2} + \frac{y^2 x^2}{2} = \sin(x) – \cos(y) + C,\)
\(\text{where } C \text{ is an integration constant.}\)
\(\text{The general solution to the differential equation is:}\)
\(\frac{y^2}{2} + \frac{y^2 x^2}{2} = \sin(x) – \cos(y) + C,\)
\(\text{expressing a relationship between } y, x, \text{ and } C. \)
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