Ever wondered how the path of a falling ball can be captured by a single equation?
💡 In Simple Words: A differential equation is a rule that links a changing quantity (like speed) to its rate of change (like acceleration). Solving it tells you the original quantity as a function of time or another variable.
What is a Differential Equation?
A differential equation (DE) is an equation that involves an unknown function and its derivatives. A derivative measures how fast something is changing, just like the speedometer tells you how quickly the car’s position changes.
Why Do We Study Differential Equations in Class 12?
- They appear in physics, biology, economics – wherever change happens.
- The ISC board expects you to recognise basic types and solve first‑order equations.
- Mastering them builds a solid base for engineering and higher‑level maths.
Basic Types of Differential Equations
- Order: the highest derivative present (first‑order, second‑order, etc.).
- Degree: the power of the highest derivative when the equation is free of radicals and fractions.
- Linear vs. Non‑linear: linear means the unknown function and its derivatives appear only to the first power and are not multiplied together.
- Separable: can be rewritten so that all y‑terms are on one side and x‑terms on the other.
How to Solve a First‑Order Separable Equation
The most common starter in the ISC exam is a separable equation. The steps are easy enough to remember with a tiny flowchart.
graph TD\nA[Identify the equation is separable] --> B[Rewrite as f(y) dy = g(x) dx] --> C[Integrate both sides] --> D[Add constant of integration] --> E[Solve for y (if required)]
Worked Example
Solve dy/dx = xy.
- Separate: dy/y = x dx.
- Integrate: ∫(1/y) dy = ∫x dx ⇒ ln|y| = x^2/2 + C.
- Exponentiate to remove ln: y = ± e^{C}·e^{x^2/2}. Combine ±e^{C} into a single constant K.
- Final solution: y = K e^{x^2/2}, where K is any real constant.
Quick Reference Table
| Step | What to Do | Why |
|---|---|---|
| 1 | Check if the equation is separable | Only separable equations can be split into y‑side and x‑side. |
| 2 | Move all y‑terms with dy, all x‑terms with dx | Prepares the equation for easy integration. |
| 3 | Integrate both sides | Integration undoes differentiation, giving the original function. |
| 4 | Add constant C | Every antiderivative has an unknown constant. |
| 5 | Solve for y (optional) | Most exam questions ask for y explicitly. |
Key Tips for the ISC Exam
- Always write the constant of integration; exam markers love it.
- If you get a ln|y| term, exponentiate to isolate y.
- Check your answer by differentiating – a quick sanity check.
- Memorise the standard forms: dy/dx = f(x)·g(y) → separable.
- Practice the small trick of swapping sides if the equation looks messy.
📝 Likely Exam Questions
- Question: Solve dy/dx = 3x^2 y.
Answer: Separate → dy/y = 3x^2 dx; integrate → ln|y| = x^3 + C; y = K e^{x^3}. - Question: Find the particular solution of dy/dx = (y/x) with y(1)=2.
Answer: Separate → dy/y = dx/x; integrate → ln|y| = ln|x| + C → y = Kx. Use y(1)=2 → K=2, so y = 2x. - Question: State the order and linearity of the equation d^2y/dx^2 + 5 dy/dx + 6y = 0.
Answer: Order = 2 (highest derivative is second); it is linear because y and its derivatives appear to the first power and are not multiplied together.
#ISC#Class 12#Mathematics#Differential Equations#Exam Tips
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