Solving the Equation dy/dx = 2 with the Initial Condition y(0) = 5

To solve the given differential equation dy/dx = 2, we can integrate both sides with respect to y.

∫(dy/dx) dx = ∫2 dy

Integrating both sides gives:

∫dy = 2∫dx

Simplifying the integrals, we have:

y = 2x + C

where C is the constant of integration.

Verifying the Solution

To verify the solution, we need to substitute the initial condition y(0) = 5 into the equation.

y(0) = 2(0) + C

5 = C

Therefore, the equation becomes:

y = 2x + 5

We can now differentiate this equation with respect to x and check if it satisfies the original differential equation.

Differentiating y = 2x + 5 with respect to x, we get:

dy/dx = 2

This is the same as the original equation dy/dx = 2, which confirms that the solution y = 2x + 5 is correct.

Explanation

The given differential equation dy/dx = 2 represents a linear function with a constant slope of 2. By integrating both sides of the equation, we find the general solution y = 2x + C, where C is the constant of integration.

To determine the specific solution, we use the initial condition y(0) = 5. Substituting this value into the equation, we solve for the constant of integration C and find C = 5.

Substituting C back into the general solution, we obtain the particular solution y = 2x + 5. To verify this solution, we differentiate it with respect to x and check if it satisfies the original differential equation.

By differentiating y = 2x + 5, we obtain dy/dx = 2, which is equal to the original equation dy/dx = 2. Therefore, the solution y = 2x + 5 is correct.

Summary

To solve the differential equation dy/dx = 2 with the initial condition y(0) = 5, we integrated both sides to find the general solution y = 2x + C. By substituting the initial condition, we determined the constant of integration as C = 5, which led to the particular solution y = 2x + 5. Finally, we verified the solution by differentiating it and confirming that it satisfies the original differential equation.