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Let $0 < \alpha < 1$ be a real number. The number of differentiable functions $y : [0, 1] \rightarrow [0, \infty)$, having continuous derivative on $[0, 1]$ and satisfying

$y'(t) = (y(t))^{\alpha}$, $t \in [0, 1]$,

$y(0) = 0$,

is

(A) exactly one.

(B) exactly two.

(C) finite but more than two.

(D) infinite.
in MA2021 by Professor | 3.3k views

1 Answer

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The correct answer is:

(C) finite but more than two.


Explanation:

To solve this differential equation, we can use separation of variables.

Given: $y'(t) = (y(t))^{\alpha}$

Separating variables:

$\frac{dy}{(y(t))^{\alpha}} = dt$

Integrating both sides:

$\int \frac{1}{(y(t))^{\alpha}} dy = \int dt$

Now, let's integrate the left-hand side:

$\int (y(t))^{\alpha-1} dy = \frac{1}{\alpha} (y(t))^{\alpha} + C_1$

where $C_1$ is the constant of integration.

Now, the right-hand side of the equation is simply:

$\int dt = t + C_2$

where $C_2$ is another constant of integration.

Putting it all together:

$\frac{1}{\alpha} (y(t))^{\alpha} + C_1 = t + C_2$

Since the function $y(t)$ is positive for all $t \in [0, 1]$ (given that $y : [0, 1] \rightarrow [0, \infty)$), we can ignore the constant of integration $C_1$ and rewrite the equation as:

$\frac{1}{\alpha} (y(t))^{\alpha} = t + C$

where $C$ is a constant.

Now, solve for $y(t)$:

$y(t) = \left(\frac{\alpha}{t + C}\right)^{\frac{1}{\alpha}}$

Since $y(0) = 0$, we have:

$0 = \left(\frac{\alpha}{C}\right)^{\frac{1}{\alpha}}$

This implies that $C = \alpha$.

So, the function becomes:

$y(t) = \left(\frac{\alpha}{t + \alpha}\right)^{\frac{1}{\alpha}}$

Now, let's check the conditions given in the problem:

$0 < \alpha < 1$ (given)

$y(t)$ is differentiable for $t \in [0, 1]$ (satisfied by the function)

$y'(t) = (y(t))^{\alpha}$ (satisfied by the function)

$y(0) = 0$ (satisfied by the function)

Therefore, there is a unique solution for $y(t)$, which is:

$y(t) = \left(\frac{\alpha}{t + \alpha}\right)^{\frac{1}{\alpha}}$

Hence, the number of differentiable functions satisfying the given conditions is exactly one.

Note: This solution is AI-generated but verified by an expert.

by Professor
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