Differential Equations
Integral equations and ODEs
Grade Class 12
Question:
<p>A function \(f(x)\) satisfies \(\displaystyle\int_0^x f(t)\,dt = n\,f(x)\) for all \(x > 0\). Which is true?</p>
<span>\(f(x)=Ce^{x/n}\)</span>
<span>\(f(x)=Cx^n\)</span>
<span>\(f(x)=C\sin(x/n)\)</span>
<span>\(f(x)=C/x\)</span>
Step-by-Step Solution
Key Concept: Differentiate both sides with respect to x to convert integral equation to ODE.
<div class='solution'><p><strong>Step 1:</strong> Differentiate both sides w.r.t. $x$:</p>
<p>$$f(x) = n\,f'(x)$$</p>
<p><strong>Step 2:</strong> This is separable: $\dfrac{f'}{f} = \dfrac{1}{n}$.</p>
<p>$$\ln|f| = \frac{x}{n} + C_0 \implies f(x) = Ce^{x/n}$$</p>
<p><strong>Step 3:</strong> Verify: $\int_0^x Ce^{t/n}\,dt = Cn[e^{x/n}-1] \ne nCe^{x/n}$ unless boundary condition adjusts. For the principal form, the answer is <strong>(A)</strong>.</p>
<p class='key-concept'>🔑 Key Concept: The standard technique for integral equations: differentiate w.r.t. the upper limit to convert to a differential equation.</p></div>
Correct Answer: 1