Let's solve the given differential equation step-by-step.
Given the differential equation:
[tex]\[ y' = 3x - 12 \][/tex]
The goal is to find the function [tex]\( y(x) \)[/tex].
Step 1: Identify the differential equation
The given equation is a first-order linear differential equation.
Step 2: Integrate both sides with respect to [tex]\( x \)[/tex]
We can find [tex]\( y(x) \)[/tex] by integrating both sides of the equation with respect to [tex]\( x \)[/tex]:
[tex]\[
\int y' \, dx = \int (3x - 12) \, dx
\][/tex]
Step 3: Solve the integral on the right-hand side
To solve the integral [tex]\( \int (3x - 12) \, dx \)[/tex], we split it into two separate integrals:
[tex]\[
\int (3x - 12) \, dx = \int 3x \, dx - \int 12 \, dx
\][/tex]
Now, integrate each term separately:
[tex]\[
\int 3x \, dx = \frac{3x^2}{2}
\][/tex]
[tex]\[
\int 12 \, dx = 12x
\][/tex]
Putting these results together, we have:
[tex]\[
\int (3x - 12) \, dx = \frac{3x^2}{2} - 12x
\][/tex]
Step 4: Include the constant of integration
Since we are solving an indefinite integral, we must include a constant of integration, denoted by [tex]\( C \)[/tex]:
[tex]\[
y(x) = \frac{3x^2}{2} - 12x + C
\][/tex]
Step 5: Write the final solution
Therefore, the general solution to the differential equation [tex]\( y' = 3x - 12 \)[/tex] is:
[tex]\[
y(x) = \frac{3x^2}{2} - 12x + C
\][/tex]
where [tex]\( C \)[/tex] is an arbitrary constant of integration.
So, our final solution can be written as:
[tex]\[
y(x) = C_1 + \frac{3x^2}{2} - 12x
\][/tex]
where [tex]\( C_1 \)[/tex] represents the integration constant.
