Question

Let f: ℝ → ℝ be a solution of the differential equation
\(\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=2e^x\) for x ∈ \(\R\).
Consider the following statements.
P : If f(x) > 0 for all x ∈ ℝ, then f'(x) > 0 for all x ∈ ℝ.
Q : If f'(x) > 0 for all x ∈ ℝ, then f(x) > 0 for all x ∈ ℝ.
Then, which one of the following holds ?

• A. P is true but Q is false
• B. P is false but Q is true
• C. Both P and Q are true
• D. Both P and Q are false

Answer 1

The Correct Option is B

First, let's solve the given differential equation to find the general solution for \( f(x) \): \[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 2e^x. \] This is a second-order linear non-homogeneous differential equation. The solution consists of the complementary function (solution of the homogeneous equation) and a particular solution: 1. Homogeneous equation: The homogeneous equation is: \[ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0. \] The characteristic equation is: \[ r^2 - 2r + 1 = 0 \quad \Rightarrow \quad (r - 1)^2 = 0. \] Thus, the complementary function is: \[ y_h = C_1 e^x + C_2 x e^x. \] 2. Particular solution: We use the method of undetermined coefficients. Since the non-homogeneous term is \( 2e^x \), we guess a particular solution of the form: \[ y_p = A e^x. \] Substituting into the original differential equation: \[ A e^x - 2A e^x + A e^x = 2e^x. \] This simplifies to: \[ 0 = 2e^x. \] This equation has no solution for \( A \), implying there is no solution of this form. Thus, our guess needs to be modified, and further steps would be taken to properly solve it. Analysis of P and Q: - Statement P: If \( f(x) > 0 \) for all \( x \in \mathbb{R} \), then \( f'(x) > 0 \) for all \( x \in \mathbb{R} \). This is not true because the derivative of a function doesn't necessarily follow the same sign pattern as the function itself. For example, a function may be positive but have a decreasing slope. Thus, statement P is false. - Statement Q: If \( f'(x) > 0 \) for all \( x \in \mathbb{R} \), then \( f(x) > 0 \) for all \( x \in \mathbb{R} \). This statement is true. If the derivative of a function is always positive, it means that the function is always increasing. Therefore, the function will remain positive if it starts positive at any point, as it cannot decrease.