Let y : (1, ∞) → R be the solution of the differential equation y'' - 2y' = 0 satisfying y(2) = 1 and lim (x→∞) y(x) = 0. Then y(3) is equal to ______ (Rounded off to two decimal p... Let y : (1, ∞) → R be the solution of the differential equation y'' - 2y' = 0 satisfying y(2) = 1 and lim (x→∞) y(x) = 0. Then y(3) is equal to ______ (Rounded off to two decimal places). Read more

Steps to Solve

1. Rewrite the differential equation

The given differential equation is

$$ y'' - 2y' = 0. $$

This is a linear homogeneous second-order differential equation.

2. Find the characteristic equation

To solve the differential equation, we first assume a solution of the form

$$ y(x) = e^{rx}. $$

Substituting into the differential equation leads to the characteristic equation:

$$ r^2 - 2r = 0. $$

3. Factor the characteristic equation

Factoring gives:

$$ r(r - 2) = 0. $$

Thus, we have the roots $r_1 = 0$ and $r_2 = 2$.

4. Write the general solution

The general solution of the differential equation is:

$$ y(x) = C_1 e^{0 \cdot x} + C_2 e^{2x} $$

which simplifies to

$$ y(x) = C_1 + C_2 e^{2x}. $$

5. Use the boundary condition y(2) = 1

Substituting $x = 2$ into the solution gives:

$$ y(2) = C_1 + C_2 e^{4} = 1. $$

6. Use the condition lim (x→∞) y(x) = 0

Since $y(x)$ has to approach 0 as $x \to \infty$, we must have $C_2 = 0$. Otherwise, the term $C_2 e^{2x}$ would grow to infinity.

This simplifies our solution to:

$$ y(x) = C_1. $$

7. Solve for C1

Substituting back gives:

$$ C_1 = 1. $$

Thus, the specific solution is:

$$ y(x) = 1. $$

8. Evaluate at y(3)

Now we can evaluate at $x = 3$:

$$ y(3) = 1. $$