Why differential equations matter in everyday life?
Ever wondered how engineers predict the cooling of a cup of tea or how biologists model population growth? Both use differential equations – the math behind change.
💡 In Simple Words: A differential equation is a recipe that tells you how a quantity changes step by step. If you know the rule, you can work out the whole story, just like following a cooking guide to bake a cake.
What is a differential equation?
A differential equation (DE) is an equation that involves a function and its derivatives. A derivative (think of it as the speed at which something is moving) shows how fast the function is changing. So a DE links a quantity with its rate of change.
First time you see these words
- Derivative: the slope or steepness of a curve at a point, like how fast a car’s speedometer moves.
- Order: the highest derivative that appears. First‑order means only the first derivative shows up.
- Linear: the function and its derivatives appear only to the first power and are not multiplied together.
Common types you’ll see in ISC exams
| Type | Form | Typical method |
|---|---|---|
| Separable | dy/dx = g(x)·h(y) | Separate variables and integrate |
| Homogeneous (first order) | dy/dx = f(y/x) | Substitute v = y/x |
| Linear (first order) | dy/dx + P(x)y = Q(x) | Integrating factor |
| Exact | M(x,y)dx + N(x,y)dy = 0 where ∂M/∂y = ∂N/∂x | Find potential function |
How to solve a first‑order separable differential equation
Most ISC starters are separable because you can pull all y‑terms to one side and all x‑terms to the other. Follow these steps:
graph TD
A[Start] --> B[Identify type]
B --> C[Separate variables]
C --> D[Integrate both sides]
D --> E[Add constant]
E --> F[Check solution]
Worked example
Solve dy/dx = 3x².
- Identify: No y on the right, so it’s already separable.
- Rewrite: dy = 3x² dx.
- Integrate: ∫dy = ∫3x² dx → y = x³ + C.
- Result: The family of curves y = x³ + C satisfies the original DE.
Notice how the constant C appears after integration – that’s the “family” of all possible solutions.
Quick checklist before you write the answer
- State the order and type of the DE.
- Separate variables correctly; don’t forget dx or dy.
- Integrate each side; add the constant of integration (C).
- Replace any substitution back to original variables.
- Verify by differentiating your final answer.
📝 Likely Exam Questions
- Question: Solve the separable DE 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 = 0.
Answer: Linear first‑order. Integrating factor = e^{∫4dx}=e^{4x}. Multiply: e^{4x}dy/dx + 4e^{4x}y = 0 → d/dx(e^{4x}y)=0 → e^{4x}y = C → y = Ce^{-4x}. - Question: Determine whether M(x,y)dx + N(x,y)dy = (y‑2x)dx + (x+3y)dy is exact, and solve if it is.
Answer: Compute ∂M/∂y = 1, ∂N/∂x = 1 → equal, so exact. Find ψ such that ψ_x = y‑2x → ψ = xy‑x² + h(y). Differentiate ψ w.r.t y: ψ_y = x + h'(y) = N = x+3y → h'(y)=3y → h = (3/2) y². Hence ψ = xy‑x² + (3/2) y² = C. - Question: A tank contains 100 L of brine with 5 kg of salt. Fresh water flows in at 2 L/min and the mixture flows out at the same rate. Write the differential equation for the amount of salt S(t) and give its solution.
Answer: Rate in = 0 (fresh water). Rate out = (2 L/min)·S/100 → dS/dt = –(2/100)S = –0.02S. Solve: S = 5 e^{–0.02t} kg.
#ISC#Class 12#Mathematics#Differential Equations#Study Notes
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