Differential Equations
Homogeneous functions — Euler's theorem
Grade Class 12
Question:
<p>If \(f(x,y)\) is a homogeneous function of degree \(n\), which is TRUE?</p>
<span>\(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf\)</span>
<span>\(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=(n-1)f\)</span>
<span>\(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=nf\)</span>
<span>\(x^2f_{xx}+y^2f_{yy}=n(n-1)f\)</span>
Step-by-Step Solution
Key Concept: Euler's theorem for homogeneous functions.
<div class='solution'><p><strong>Euler's Theorem:</strong> If \(f(x,y)\) is homogeneous of degree \(n\), then:</p>
<p>\[x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf\]</p>
<p><strong>Proof sketch:</strong> Since \(f(tx,ty) = t^n f(x,y)\), differentiate both sides w.r.t. \(t\) and set \(t=1\):</p>
<p>\[x f_x(tx,ty) + y f_y(tx,ty) = nt^{n-1}f(x,y)\]</p>
<p>At \(t=1\): \(xf_x + yf_y = nf\). <strong>Answer: (A)</strong>.</p>
<p>Note: Option (D) \(x^2 f_{xx} + 2xyf_{xy} + y^2 f_{yy} = n(n-1)f\) is the second-order Euler theorem (not exactly option D as stated).</p>
<p class='key-concept'>🔑 Key Concept: Euler's theorem is fundamental. Memorise: \(x f_x + y f_y = nf\) for degree-\(n\) homogeneous functions.</p></div>
Correct Answer: 1