Question

General solution of $(D^2 - 5D + 6)y = 0$ is $y(x) =$

• A. $c_1 e^{-3x} + c_2 e^{2x}$
• B. $c_1 e^{3x} + c_2 e^{-2x}$
• C. $c_1 e^{3x} + c_2 e^{2x}$
• D. $c_1 e^{-3x} + c_2 e^{-2x}$

Hint:

For a second-order linear homogeneous differential equation, solve the characteristic equation; distinct real roots \( r_1, r_2 \) give the solution \( c_1 e^{r_1 x} + c_2 e^{r_2 x} \).

Answer 1

The Correct Option is C

Step 1: Write the characteristic equation.
The given differential equation is: \[ (D^2 - 5D + 6)y = 0, \] where \( D = \frac{d}{dx} \), so it can be rewritten as: \[ \frac{d^2y}{dx^2} - 5 \frac{dy}{dx} + 6y = 0. \] This is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation is: \[ r^2 - 5r + 6 = 0. \] Step 2: Solve the characteristic equation.
Solve the quadratic equation: \[ r^2 - 5r + 6 = 0, \] using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -5 \), \( c = 6 \): \[ r = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}. \] \[ r_1 = \frac{5 + 1}{2} = 3, \quad r_2 = \frac{5 - 1}{2} = 2. \] The roots are \( r = 3 \) and \( r = 2 \), which are real and distinct. Step 3: Write the general solution.
For a second-order linear homogeneous differential equation with distinct real roots \( r_1 \) and \( r_2 \), the general solution is: \[ y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}. \] Substituting the roots \( r_1 = 3 \) and \( r_2 = 2 \): \[ y(x) = c_1 e^{3x} + c_2 e^{2x}. \] Step 4: Evaluate the options.
(1) \( c_1 e^{-3x} + c_2 e^{2x} \): Incorrect, as the roots are 3 and 2, not \(-3\) and 2. Incorrect.
(2) \( c_1 e^{3x} + c_2 e^{-2x} \): Incorrect, as the roots are 3 and 2, not 3 and \(-2\). Incorrect.
(3) \( c_1 e^{3x} + c_2 e^{2x} \): Correct, as this matches the general solution with roots 3 and 2. Correct.
(4) \( c_1 e^{-3x} + c_2 e^{-2x} \): Incorrect, as the roots are 3 and 2, not \(-3\) and \(-2\). Incorrect.
Step 5: Select the correct answer.
The general solution is \( y(x) = c_1 e^{3x} + c_2 e^{2x} \), matching option (3).