Question

The solution of the differential equation \( \frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + 2y = 0 \) is given by

• A. \( y = C_1 e^x + C_2 e^{2x} \)
• B. \( y = C_1 e^x + C_2 e^{-2x} \)
• C. \( y = C_1 e^x + C_2 e^{2x} \)
• D. \( y = C_1 e^x + C_2 e^{2x} \)

Hint:

When solving second-order linear differential equations with constant coefficients, always find the characteristic equation by assuming a solution of the form \( y = e^{rx} \), then solve for the roots of the characteristic equation.

Answer 1

The Correct Option is C

Step 1: Understanding the differential equation.
We are given the second-order linear homogeneous differential equation: \[ \frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + 2y = 0 \] This is a linear differential equation with constant coefficients. To solve this type of equation, we first write down the characteristic equation and solve for the roots.
Step 2: Finding the characteristic equation.
Assume that the solution is of the form \( y = e^{rx} \), where \( r \) is a constant. Then, we can calculate the first and second derivatives: \[ \frac{dy}{dx} = r e^{rx}, \quad \frac{d^2 y}{dx^2} = r^2 e^{rx} \] Substituting these into the given equation: \[ r^2 e^{rx} - 3r e^{rx} + 2 e^{rx} = 0 \] Factor out \( e^{rx} \) (which is never zero): \[ e^{rx} (r^2 - 3r + 2) = 0 \] Thus, the characteristic equation is: \[ r^2 - 3r + 2 = 0 \]
Step 3: Solving the characteristic equation.
Now, solve the quadratic equation \( r^2 - 3r + 2 = 0 \). Factorizing: \[ (r - 1)(r - 2) = 0 \] So, the roots are: \[ r_1 = 1 \quad \text{and} \quad r_2 = 2 \]
Step 4: General solution.
Since the roots \( r_1 = 1 \) and \( r_2 = 2 \) are real and distinct, the general solution to the differential equation is: \[ y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} \] Substituting the values of \( r_1 \) and \( r_2 \): \[ y(x) = C_1 e^x + C_2 e^{2x} \]
Step 5: Conclusion.
The general solution to the given differential equation is: \[ y(x) = C_1 e^x + C_2 e^{2x} \] Therefore, the correct answer is: \[ \boxed{\text{(C) } y = C_1 e^x + C_2 e^{2x}} \]

Answer 2

Another way to confirm the general solution is to substitute each candidate function directly into the differential equation \( \dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} + 2y = 0 \) and check whether it is satisfied identically, rather than deriving the characteristic equation from scratch.

1. \( y = C_1e^{x} + C_2e^{2x} \) (first listed form): Differentiating twice gives \( y' = C_1e^{x}+2C_2e^{2x} \) and \( y'' = C_1e^{x}+4C_2e^{2x} \). Substituting: \( (C_1e^{x}+4C_2e^{2x}) - 3(C_1e^{x}+2C_2e^{2x}) + 2(C_1e^{x}+C_2e^{2x}) = C_1e^{x}(1-3+2) + C_2e^{2x}(4-6+2) = 0 \), so this form satisfies the equation for any \( C_1, C_2 \).
2. \( y = C_1e^{x} + C_2e^{-2x} \): Differentiating gives \( y' = C_1e^{x}-2C_2e^{-2x} \) and \( y'' = C_1e^{x}+4C_2e^{-2x} \). Substituting: \( (C_1e^{x}+4C_2e^{-2x}) - 3(C_1e^{x}-2C_2e^{-2x}) + 2(C_1e^{x}+C_2e^{-2x}) = C_1e^{x}(1-3+2) + C_2e^{-2x}(4+6+2) = 12C_2e^{-2x} \), which is not identically zero unless \( C_2 = 0 \), so this form does not satisfy the equation in general.
3. \( y = C_1e^{x} + C_2e^{2x} \) (repeated as the third option): Identical in form to the first option, so it satisfies the equation by the same substitution shown above.
4. \( y = C_1e^{x} + C_2e^{2x} \) (repeated as the fourth option): Also identical in form to the first option, and likewise satisfies the equation.

Direct substitution confirms that only the form \( y = C_1e^{x}+C_2e^{2x} \), appearing as three of the four listed options, actually satisfies the differential equation, while the exponent \( e^{-2x} \) form does not.

Therefore, the correct answer is \( y = C_1e^{x} + C_2e^{2x} \).