Problem 1 - Essential Calculus

Question

Solve the differential equation. $$ \frac{d y}{d x}=x+2 $$

Step-by-Step Solution

Answer

The solution to the differential equation is \(y = \frac{1}{2}x^2 + 2x + C\), where \(C\) is a constant.

Step 1: Label the equation

The given differential equation is \(\frac{d y}{d x} = x+2\). We want to find the function \(y\) that satisfies this equation.

Step 2: Separation of variables

We start by relocating terms of the same variable on the same side of the equation. In this case, this means moving \(dx\) to the other side to obtain: \(dy = (x+2) dx\).

Step 3: Integrate both sides

In order to find \(y\), we need to integrate both sides of the equation. The integration of \(dy\) is \(y\), and the integration of \((x+2)dx\) is \(\frac{1}{2}x^2 + 2x\). So, our equation becomes: \(y = \frac{1}{2}x^2 + 2x + C\), where \(C\) is the constant of integration.