Question

A function $y(x)$ such that $y(0) = 1$ and $y(1) = 4e$ is a solution of the differential equation $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0$. Then, $y(2)$ is equal to _______}

• A. $-5e^2$
• B. $7e^2$
• C. $-5e$
• D. $7e$

Hint:

For repeated roots $m$, the general solution always takes the form $y(x) = (C_1 + C_2 x)e^{mx}$.
Be careful not to forget the $x$ multiplier in the second term.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a second-order linear homogeneous differential equation with constant coefficients:
\[ \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0 \] We need to find the specific solution $y(x)$ using the boundary conditions $y(0) = 1$ and $y(1) = 4e$, and then evaluate $y(2)$.

Step 2: Key Formula or Approach:
The auxiliary (characteristic) equation is:
\[ m^2 - 2m + 1 = 0 \] Solve this quadratic equation to find the roots and write down the general solution.

Step 3: Detailed Explanation:

• Factor the auxiliary equation:
\[ (m - 1)^2 = 0 \implies m = 1, 1 \]
• Since we have real and repeated roots, the general solution is:
\[ y(x) = (C_1 + C_2 x) e^x \]
• Apply the first boundary condition $y(0) = 1$:
\[ y(0) = (C_1 + C_2(0)) e^0 = C_1 = 1 \]
• Apply the second boundary condition $y(1) = 4e$:
\[ y(1) = (C_1 + C_2(1)) e^1 = (1 + C_2) e = 4e \]
• Divide both sides by $e$ (since $e \neq 0$):
\[ 1 + C_2 = 4 \implies C_2 = 3 \]
• Write down the unique particular solution:
\[ y(x) = (1 + 3x) e^x \]
• Calculate the value of $y(x)$ at $x = 2$:
\[ y(2) = (1 + 3(2)) e^2 = (1 + 6) e^2 = 7e^2 \]

Step 4: Final Answer:
The value of $y(2)$ is $7e^2$.