Separation of Variables

• A separable differential equation is a first-order equation that can be expressed in the form $\frac{dy}{dx} = f(x)g(y)$, where the rate of change is a product of two independent functions of each variable.

• The process involves treating the derivative $\frac{dy}{dx}$ as a ratio of differentials, enabling the algebraic 'separation' of the $y$ terms and $x$ terms to opposite sides of the equation.

• The General Solution represents a whole family of curves and includes an arbitrary constant of integration, $c$, which accounts for all possible functions that satisfy the derivative relationship.

Step-by-Step Methodology

• Step 1: Rearrange: Move all terms containing $y$ (including $dy$) to one side and all terms containing $x$ (including $dx$) to the other using multiplication or division.
• Step 2: Integrate: Apply integration rules to both sides independently, ensuring that the integral of the reciprocal $1/g(y)$ is handled correctly.
• Step 3: Constant of Integration: Add $+c$ immediately after integrating; this is the most critical step for preserving the general nature of the solution.
• Step 4: Boundary Conditions: If specific values are provided (e.g., $y=a$ when $x=b$), substitute them into the integrated equation to calculate the precise value of $c$.
• Step 5: Isolate y: Whenever possible, use algebraic manipulation (like exponentiation or taking logarithms) to express the solution in the explicit form $y = f(x)$.

• General vs. Particular Solutions: A general solution describes the behavior of a system generally, while a particular solution uses a specific point to pin down one exact trajectory out of the infinite possibilities.

• Separable vs. Non-separable: A separable equation must be reducible to a product $f(x) \cdot g(y)$; equations involving sums of variables like $\frac{dy}{dx} = x + y$ cannot be solved using this specific technique.

• Logarithmic Simplification: If the integration results in $\ln |y| = f(x) + c$, it is often more efficient to write the constant as $\ln |A|$, allowing you to use log laws to reach $y = Ae^{f(x)}$.

• Verify the Algebra: Before integrating, double-check that your $dy$ and $dx$ terms are in the 'numerator' position; if a differential is in the denominator, your integration will be invalid.

• Sanity Check: For modelling questions, check if your final particular solution makes sense at the boundary; for example, if $t=0$, does your formula yield the initial value given in the prompt?

• Delayed Constant Addition: Many students wait until the final step to add $+c$, but the constant must be added during the integration step to ensure it is correctly affected by subsequent operations like exponentiation.

• Incorrect Reciprocals: When moving $g(y)$ to the left side, it must become $1/g(y)$; failing to use the reciprocal is a frequent algebraic error that leads to an entirely different solution family.

• Missing Modulus: When integrating $1/y$, remember that $\int \frac{1}{y} dy = \ln |y|$; ignoring the absolute value can lead to missing parts of the solution domain.