Separation of Variables

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

• The goal of the method is to isolate all terms involving $y$ (including $dy$) on one side and all terms involving $x$ (including $dx$) on the other.

• Once separated, the equation takes the form $\frac{1}{g(y)} dy = f(x) dx$, which can then be solved using standard integration techniques.

Step-by-Step Methodology

• Step 1: Rearrange. Move all $y$ terms to the left with $dy$ and all $x$ terms to the right with $dx$. Ensure $dy$ and $dx$ are in the numerators.
• Step 2: Integrate. Apply integration to both sides of the equation: $\int \frac{1}{g(y)} dy = \int f(x) dx$
• Step 3: Add Constant. Include the constant $+C$ immediately after integrating. This is the most common point of failure in student work.
• Step 4: Solve for y. If possible, rearrange the resulting equation to express $y$ as an explicit function of $x$. This is the General Solution.
• Step 5: Apply Conditions. If an initial condition (e.g., $y=y_0$ when $x=x_0$) is provided, substitute these values to find the specific value of $C$. This results in the Particular Solution.

• Check for Separability: Before starting, ensure the equation is a product/quotient. If you see terms like $\frac{dy}{dx} = x + y$, the variables cannot be separated by multiplication/division, and this method will not work.

• The Exponential Trap: If your integration results in $\ln|y| = f(x) + C$, remember that $y = e^{f(x) + C} = e^C \cdot e^{f(x)}$. It is standard practice to replace $e^C$ with a new constant $A$, giving $y = Ae^{f(x)}$.

• Verify the Solution: You can always check your answer by differentiating your result. If $\frac{dy}{dx}$ of your solution matches the original differential equation, your work is correct.

• Units and Context: In modelling questions, ensure your variables match the context (e.g., $t$ for time, $P$ for population) and that your final answer is in a sensible format.

• Forgetting $+C$: Adding the constant at the very end of the algebra instead of immediately after integration leads to incorrect particular solutions.

• Illegal Algebra: Attempting to separate variables by adding or subtracting terms across the equals sign (e.g., moving $y$ from $\frac{dy}{dx} = x + y$ by subtracting $y$) is a common error; separation must be done via multiplication or division.

• Logarithm Errors: Forgetting the absolute value bars in $\int \frac{1}{y} dy = \ln|y|$ can lead to lost solutions, especially when dealing with negative initial conditions.