Questions: Solve the initial value problem: dy/dx + 6y = 9, y(0)=0 y(x)=

Solution Steps

To solve the initial value problem, we need to solve the first-order linear differential equation \(\frac{d y}{d x} + 6y = 9\) with the initial condition \(y(0) = 0\). We can use the integrating factor method to find the general solution and then apply the initial condition to find the particular solution.

We start with the given first-order linear differential equation: \[ \frac{d y}{d x} + 6y = 9 \] with the initial condition: \[ y(0) = 0 \]

To solve this differential equation, we use the integrating factor method. The integrating factor \( \mu(x) \) is given by: \[ \mu(x) = e^{\int 6 \, dx} = e^{6x} \]

Multiplying both sides of the differential equation by the integrating factor: \[ e^{6x} \frac{d y}{d x} + 6 e^{6x} y = 9 e^{6x} \]

This simplifies to: \[ \frac{d}{dx} \left( e^{6x} y \right) = 9 e^{6x} \]

Integrating both sides with respect to \( x \): \[ e^{6x} y = \int 9 e^{6x} \, dx = \frac{9}{6} e^{6x} + C = \frac{3}{2} e^{6x} + C \]

Solving for \( y \): \[ y = \frac{3}{2} + Ce^{-6x} \]

We apply the initial condition \( y(0) = 0 \): \[ 0 = \frac{3}{2} + C e^{0} \implies 0 = \frac{3}{2} + C \implies C = -\frac{3}{2} \]

Final Answer