Q.At any point $(x, y)$ of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point $(-4, -3)$. Find the equation of the curve given that it passes through $(-2, 1)$.

Differential Equation Modeling: From Intuition to Precision

Imagine you're watching a cup of hot coffee cool down on your desk. It starts hot, then gradually loses heat to the room. But here's the key question: how fast does it cool at any given moment? The answer isn't a fixed number — it depends on how hot the coffee is right now. The hotter it is, the faster it cools. This is the core idea behind differential equation modeling: the rate of change of a quantity depends on the quantity itself.

The Intuition First

Suppose the room is at $20^\circ$C and your coffee starts at $90^\circ$C. Newton's Law of Cooling says the rate at which the coffee cools is proportional to the temperature difference between the coffee and the room. So when the coffee is $90^\circ$C, the difference is $70^\circ$C and it cools fast; when it's $40^\circ$C, the difference is only $20^\circ$C and it cools slowly. The rate changes as the temperature changes.

If we let $T(t)$ be the temperature at time $t$, then "rate of change of temperature" is $\frac{dT}{dt}$, and the statement above becomes:

The minus sign is because the temperature is decreasing; the constant $k$ depends on the cup, the liquid, etc. This is a differential equation — an equation that involves a function and its derivative.

The Precise Statement

A differential equation is any equation that contains an unknown function and one or more of its derivatives. The goal is to find the function itself (here, $T(t)$) that satisfies the equation.

Modeling means taking a real-world situation and translating it into a differential equation:

1. Identify the quantity you want to study (population, temperature, drug concentration, voltage).
2. State the rate law in words: "The population grows at a rate proportional to its size."
3. Translate into math: $\frac{dP}{dt} = kP$.
4. Add initial conditions: $P(0) = P_0$ (the starting value).

Why This Matters

Without differential equations, you'd have to guess the whole future behavior of a system. With them, you get a precise rule that governs every instant of change. The solution to $\frac{dP}{dt} = kP$ is $P(t) = P_0 e^{kt}$ — exponential growth. The solution to the cooling equation is $T(t) = 20 + 70 e^{-kt}$ — exponential decay toward room temperature.

A Simple Example to Try

A bacteria culture doubles every hour. If you start with 100 bacteria:

Step 1: Quantity = population $P(t)$. …