Why care about differential equations?
Ever wondered how engineers predict a car's speed or how biologists model population growth? They both use differential equations – the math language of change.
💡 In Simple Words: A differential equation is a rule that tells you how something is changing at each instant. If you know the rule, you can work backwards to find the original quantity.
What is a differential equation?
A differential equation (or DE) is an equation that involves a function and its derivative. A derivative measures how fast something is changing – think of it like the speedometer of a car, while the function is the distance travelled.
Key parts to spot
- Function (y): the unknown thing we want to find.
- Derivative (dy/dx): the rate at which y changes with respect to x.
- Equation: ties the two together, often with x and constants.
First‑order differential equations
These involve only the first derivative (dy/dx). They are the easiest to solve and show up a lot in ISC exams.
Common types
| Type | Form | How we solve it |
|---|---|---|
| Separable | dy/dx = g(x)·h(y) | Move all y‑terms to one side, x‑terms to the other, then integrate. |
| Linear | dy/dx + P(x)·y = Q(x) | Use an integrating factor = e^(∫P(x)dx) to turn it into a product. |
| Exact | M(x,y)dx + N(x,y)dy = 0 where ∂M/∂y = ∂N/∂x | Find a potential function ψ(x,y) whose partial derivatives match M and N. |
Solving a separable DE – step by step
Let’s walk through a classic example that often appears in the ISC board:
Example: Solve dy/dx = 3x²·y.
Follow these steps:
Step 1 – Identify Equation: The derivative dy/dx is multiplied by a function of x and y, so it looks separable.
Step 2 – Separate variables: Rewrite as dy/y = 3x² dx.
Step 3 – Integrate: ∫(1/y)dy = ∫3x²dx → ln|y| = x³ + C₁.
Step 4 – Solve for y: Exponentiate both sides: y = C·e^{x³}, where C = e^{C₁}.
That’s the whole solution! Notice how each step follows a clear pattern – that’s why the flowchart helps.
Quick checklist for solving first‑order DEs
- Is the equation separable? If yes, move terms and integrate.
- If not, check if it fits the linear form dy/dx + P(x)y = Q(x). Use the integrating factor.
- Otherwise, test the exactness condition ∂M/∂y = ∂N/∂x.
- Never forget the constant of integration – it’s what makes the answer a family of curves.
Why the constant matters
Imagine you’re drawing a family of curves that all satisfy the same rule. The constant C slides you along that family, just like moving a ruler up or down changes the line’s intercept.
Typical pitfalls and how to avoid them
- Dropping the constant: Always write + C after integration.
- Mixing up variables: Keep y‑terms on the left, x‑terms on the right when separating.
- Wrong integrating factor: Remember it’s e^(∫P(x)dx), not just P(x).
📝 Likely Exam Questions
- Question: Solve dy/dx = (2x)/(y).
Answer: Separate: y dy = 2x dx → ∫y dy = ∫2x dx → (1/2) y² = x² + C → y² = 2x² + C'. - Question: Find the general solution of dy/dx + 4y = 8e^{2x}.
Answer: Linear form with P=4, Q=8e^{2x}. Integrating factor = e^{∫4dx}=e^{4x}. Multiply: e^{4x}dy/dx + 4e^{4x}y = 8e^{6x}. Left side = d/dx(e^{4x}y). Integrate: e^{4x}y = ∫8e^{6x}dx = (8/6)e^{6x}+C = (4/3)e^{6x}+C. Hence y = (4/3)e^{2x}+Ce^{-4x}. - Question: Determine whether M(x,y)dx + N(x,y)dy = 0 with M=2xy and N=x² is exact, and solve if it is.
Answer: Compute ∂M/∂y = 2x, ∂N/∂x = 2x → equal, so exact. Find ψ such that ψ_x = 2xy → ψ = x²y + h(y). Differentiate ψ wrt y: ψ_y = x² + h'(y) = N = x² → h'(y)=0 → h = C. Thus ψ = x²y = C → solution: x²y = C. - Question: Explain in one sentence why the constant of integration cannot be omitted. Answer: Without + C, you’d get only a single curve instead of the whole family of solutions that satisfy the differential equation.