Question 15: Find the equation of a curve passing through the point (0, 0) and whose differential equation is y' = e^ extsin x.

In one line

The rule

To solve a differential equation of the form dy/dx = f(x), integrate both sides and use the given point to find the constant.

Formula: dy/dx = f(x) ⇒ y = ∫f(x)dx + C Here, f(x) = e^{sin x}.

Let's understand it simply

Imagine you are walking along a path. At every step, the slope (steepness) of the path is not fixed, but is given by e^(sin x). This means, as you move along the x-axis, the way the path rises or falls depends on the value of sin x at that point. You want to find the exact shape of this path, and you know it passes through the starting point (0, 0).

How it works

1. Start with the differential equation: dy/dx = e^{sin x}.
2. To find y, integrate both sides with respect to x.
3. This gives y = ∫e^{sin x} dx + C.
4. Use the given point (0, 0) to find the value of C.
5. Write the final equation of the curve.

Worked example

Given: dy/dx = e^{sin x}, passes through (0, 0)
Step 1: Integrate both sides: y = ∫e^{sin x} dx + C
Step 2: Let’s keep the integral as it is (since ∫e^{sin x} dx has no simple form)
Step 3: Use the point (0, 0):
        When x = 0, y = 0
        0 = ∫e^{sin x} dx (from 0 to 0) + C
        0 = 0 + C
        So, C = 0
Step 4: Final answer: y = ∫e^{sin x} dx (from 0 to x)

Or, you can write: y(x) = ∫₀ˣ e^{sin t} dt

Where you see it in real life

• In physics, when a changing rate (like speed) depends on a sine function, such as a pendulum's motion.
• In engineering, for signals that vary with sine waves, like alternating current (AC).
• In biology, to model growth rates that change in a wave-like pattern.
• In finance, for interest rates that fluctuate periodically.

Why it matters

• Solving such equations helps you predict how things change over time.
• It is the basis for understanding many natural and man-made systems.
• This skill is needed for higher studies in maths, physics, and engineering.

Common mistakes

• Forgetting to add the constant C after integrating.
• Not using the given point to find C.
• Trying to write the integral as a simple formula (here, it does not have one).

Remember it

When you see dy/dx = f(x), integrate f(x), add C, and use the given point to find C.