Question

Let \( y' \) and \( y'' \) denote the first and second derivatives of \( y \). Let \( y(x) \) satisfy:
\[ y'' - 3y' + 2y = 0,\quad y(0)=1,\quad y'(0)=3. \]
Then \(y''(0)\) (in integer) is ____________.

Hint:

For linear ODEs, you can directly substitute initial conditions into the rearranged equation to find unknown derivatives.

Answer 1

Correct Answer: 7

We use the differential equation:
\[ y'' - 3y' + 2y = 0 \]
Rearranging gives:
\[ y'' = 3y' - 2y \]
Now substitute the initial values at \(x=0\):
\(y(0)=1\)
\(y'(0)=3\)
So,
\[ y''(0) = 3(3) - 2(1) \]
\[ = 9 - 2 = 7 \]
Thus, \(y''(0)=7\).
Final Answer: 7