Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
If the solution of the differential equation $\frac{dy}{dx} - y = 1 - e^{-x}$ and $y(0) = y_0$ has a finite value, when $x \to \infty$, then the value of $|2/y_0|$ is ______.
Step-by-Step Solution
Key Concept: For linear first-order ODEs, multiply by the integrating factor $e^{\int P dx}$ to convert the left side to a derivative.
The linear first-order differential equation $\frac{dy}{dx} - y = 1 - e^{-x}$ has integrating factor $I.F. = e^{-x}$. Multiplying through gives $ye^{-x} = \int e^{-x}(1 - e^{-x})dx = -e^{-x} + \frac{1}{2}e^{-2x} + C$. Solving for $y$: $y = -1 + \frac{1}{2}e^{-x} + Ce^x$. Using the boundary condition that as $x \to \infty$, $y$ approaches a finite value requires $C = 0$, and the initial condition $y_0 + \frac{1}{2} = 0$ gives $y_0 = -\frac{1}{2}$.
Correct Answer: 4