Question

The general solution for the differential equation `dy/dx = e^(3x - y)` is ______.

पर्याय

• 3e^y = e^3x + C

• log (3x − y) = C

• `e^(3x - y) = C`

• −e^y + 3e^3x = C

Answer 1

The general solution for the differential equation `dy/dx = e^(3x - y)` is 3e^y = e^3x + C.

Explanation:

Using the property of exponents `(e^(a - b) = e^a/e^b)`, we can rewrite the equation as:

`dy/dx (e^3x)/(e^y)`

e^y dy = e^3x dx

Integrate the left side with respect to y and the right side with respect to x:

`int e^y dy = inte^(3x) dx`

The integral of e^y is e^y

The integral of e^3x is `e^(3x)/3`   ...(using the substitution u = 3x).

so, we get:

e^y = `e^(3x)/3 + C'`

Simplify the Equation

3e^y = e^3x + 3C′   ...(multiply by 3)

Since 3C′ is just another constant, we can replace it with C:

3e^y = e^3x + C