Question

Solve the initial value problem: dx/dx +6y= 4, y(0) = 0
y(x)=__.

Answer 1

To solve the initial value problem dx/dx + 6y = 4 with the initial condition y(0) = 0, integrate both sides of the equation and use the initial condition to find the integration constants.

Step-by-step explanation:

To solve the initial value problem, we need to find the function y(x) that satisfies the given equation dx/dx + 6y = 4 and the initial condition y(0) = 0.

We can solve this first-order linear differential equation by multiplying both sides by dx and integrating:

dx/dx + 6y = 4

Integrating both sides with respect to x gives:

x + 6∫y dy = ∫4 dx

(x + 3y²) / 2 + C₁ = 4x + C₂

Now we can use the initial condition y(0) = 0 to find the values of the integration constants C₁ and C₂:

(0 + 3(0)²) / 2 + C₁ = 4(0) + C₂

C₁ = C₂

Therefore, the solution to the initial value problem is:

(x + 3y²) / 2 = 4x + C