Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)
Question Statement:
Consider the initial value problem $\frac{dy}{dx} + 2y = 4$ with $y(0) = 1$. As $x \to \infty$, the limiting value of $y(x)$ is equal to ________ (answer in integer).
Answer Key:
2
Step-by-Step Solution & Explanation
This is a linear ODE with integrating factor $IF = e^{\int 2 dx} = e^{2x}$.
Multiplying both sides by $e^{2x}$:
$\frac{d}{dx}(y e^{2x}) = 4 e^{2x}$
Integrating both sides:
$y e^{2x} = \int 4 e^{2x} dx = 2 e^{2x} + C \implies y(x) = 2 + C e^{-2x}$
Apply the initial condition $y(0) = 1$:
$1 = 2 + C \implies C = -1$
So the exact solution is:
$y(x) = 2 - e^{-2x}$
Taking the limit as $x \to \infty$:
$\lim_{x \to \infty} y(x) = 2 - e^{-\infty} = 2 - 0 = 2$