Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
If $f(x), g(x)$ be twice differentiable functions on [0, 2] satisfying $f''(x) = g''(x)$, $f'(1) = 2g'(1) = 4$ and $f(2) = 3g(2) = 9$, then:
$f(4) - g(4) = 10$
$|f(x) - g(x)| < 2 \Rightarrow -2 < x < 0$
$f(2) - g(2) = x = -1$
$f(x) - g(x) = 2x$ has real root
Step-by-Step Solution
Key Concept: Successive integration of the second derivative condition with initial values determines both constants uniquely.
Given $f''(x) = g''(x)$, integrate to get $f'(x) = g'(x) + c$. Using $f'(1) = 4$ and $g'(1) = 2$, we find $c = 2$. Integrate again to get $f(x) = g(x) + cx + d$. Using $f(2) = 9$ and $g(2) = 3$ with $c = 2$, we find $d = 2$. Therefore $f(x) - g(x) = 2x + 2$.
Correct Answer: 1,2