Question

If \(y=Ae^{mx}+Be^{nx}\), then show that \(\dfrac{d^2y}{dx^2}-(m+n)\dfrac{dy}{dx}+mny=0\).

Hint:

Differentiate twice, substitute into the given expression, and show the coefficients of e^mx and e^nx each vanish.

Answer 1

Step 1: Find the first derivative:
\[ y'=Am\,e^{mx}+Bn\,e^{nx} \]

Step 2: Find the second derivative:
\[ y''=Am^2e^{mx}+Bn^2e^{nx} \]

Step 3: Substitute into \(y''-(m+n)y'+mny\) and group by \(e^{mx}\) and \(e^{nx}\):
Coefficient of \(Ae^{mx}\): \(m^2-(m+n)m+mn=m^2-m^2-mn+mn=0\).
Coefficient of \(Be^{nx}\): \(n^2-(m+n)n+mn=n^2-mn-n^2+mn=0\).

Step 4: Both coefficients vanish, so the whole expression is zero:
\[ y''-(m+n)y'+mny=0\cdot Ae^{mx}+0\cdot Be^{nx}=0 \]

Final Answer:
\[ \boxed{\dfrac{d^2y}{dx^2}-(m+n)\dfrac{dy}{dx}+mny=0} \]