The given differential equation is:

$y' - 2y = 3$

Step 1: Solve the Differential Equation

This is a first-order linear differential equation of the form:

$y' - 2y = 3$

Find the Integrating Factor (IF)

The standard form is:

$y' + P(t)y = Q(t)$

where $P(t) = -2$ and $Q(t) = 3$.

The integrating factor is:

$\mu(t) = e^{\int -2 dt} = e^{-2t}$

Multiply the equation by $e^{-2t}$:

$e^{-2t} y' - 2e^{-2t} y = 3e^{-2t}$

Recognizing the left-hand side as the derivative of $y e^{-2t}$, we rewrite:

$\frac{d}{dt} \left( y e^{-2t} \right) = 3e^{-2t}$

Integrate Both Sides

$\int \frac{d}{dt} (y e^{-2t}) dt = \int 3e^{-2t} dt$

$y e^{-2t} = \int 3e^{-2t} dt$

$y e^{-2t} = \frac{3}{-2} e^{-2t} + C$

$y e^{-2t} = -\frac{3}{2} e^{-2t} + C$

$y = Ce^{2t} - \frac{3}{2}$

Step 2: Behavior as $t \to \infty$

• If $C > 0$, $y \to \infty$ as $t \to \infty$.
• If $C < 0$, $y \to -\infty$ as $t \to \infty$.
• If $C = 0$, $y$ approaches $-\frac{3}{2}$.

This means that in general, solutions grow exponentially unless $C = 0$, in which case they approach $-\frac{3}{2}$.

Next Steps:

1. Would you like a direction field plot for visualization?
2. Do you need more clarification on any of the steps?
3. Do you want help with initial conditions?

Related Questions

1. How do you determine the integrating factor for any first-order linear differential equation?
2. What is the significance of the constant $C$ in the general solution?
3. How does the behavior of solutions change when modifying the coefficient of $y$ in the equation?
4. How can a direction field help in understanding solution behavior?
5. What happens if the differential equation is homogeneous (i.e., $Q(t) = 0$)?

Tip:

For non-homogeneous linear ODEs like this, always solve the homogeneous part first and then find a particular solution!