Question

The differential equation among the following, whose general solution is \( y=A e^{5x}+B e^{-4x} \) is

• A. \( 5\frac{dy}{dx}+4\frac{dx}{dy}=0 \)
• B. \( \frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}+20y=0 \)
• C. \( \left(\frac{dy}{dx}\right)^{2}-\frac{dy}{dx}-20y=0 \)
• D. \( \frac{d^{2}y}{dx^{2}}-\frac{dy}{dx}-20y=0 \)

Hint:

The middle term coefficient of the second-order differential equation is always equal to \( -(\text{sum of roots}) \), and the final constant multiplier is equal to the \( (\text{product of roots}) \). This simple rule allows you to check options mentally in seconds!

Answer 1

The Correct Option is D

Step 1: Read the roots from the solution.
The solution $y=Ae^{5x}+Be^{-4x}$ comes from a characteristic equation whose roots are the powers of $e$. So $m_1=5$ and $m_2=-4$.
Step 2: Recall how the roots build the equation.
If the roots are $m_1,m_2$, the characteristic equation is \[ m^2-(m_1+m_2)m+m_1m_2=0 \]
Step 3: Find the sum of roots.
\[ m_1+m_2=5+(-4)=1 \]
Step 4: Find the product of roots.
\[ m_1m_2=5\times(-4)=-20 \]
Step 5: Write the characteristic equation.
\[ m^2-(1)m+(-20)=0\implies m^2-m-20=0 \]
Step 6: Convert to derivative form.
Replace $m^2$ by $\frac{d^2y}{dx^2}$ and $m$ by $\frac{dy}{dx}$: \[ \frac{d^2y}{dx^2}-\frac{dy}{dx}-20y=0 \] \[ \boxed{\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}-20y=0} \]