Differential Equations
First Order Linear ODEs
Grade 12
Question:
<p>If \(y_1(x)\) and \(y_2(x)\) are two solutions of \(\frac{dy}{dx} + f(x)y = r(x)\), then \(y_1(x) + y_2(x)\) is solution of</p>
<p>(A) \(\frac{dy}{dx} + f(x)y = 0\)</p>
<p>(B) \(\frac{dy}{dx} + 2f(x)y = r(x)\)</p>
<p>(C) \(\frac{dy}{dx} + f(x)y = 2r(x)\)</p>
<p>(D) \(\frac{dy}{dx} + 2f(x)y = 2r(x)\)</p>
Step-by-Step Solution
Key Concept: Use the superposition principle: if both are solutions to the linear ODE, their sum satisfies a modified equation.
<p>If $y_1$ and $y_2$ are solutions of $\frac{dy}{dx} + f(x)y = r(x)$, then $\frac{dy_1}{dx} + f(x)y_1 = r(x)$ and $\frac{dy_2}{dx} + f(x)y_2 = r(x)$. Adding these: $\frac{d(y_1 + y_2)}{dx} + f(x)(y_1 + y_2) = 2r(x)$.</p>
Correct Answer: C