The solution of the differential equation $$ \frac{d^{2} y}{d t^{2}}-y=0 \text {, } $$ subject to the boundary conditions $$ y(0)=1 $$ and $$ y(\infty)=0 $$, is

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To solve the differential equation

$$
\frac{d^{2} y}{d t^{2}} - y = 0
$$

with the boundary conditions $$ y(0) = 1 $$ and $$ y(\infty) = 0 $$, follow these steps:

1. **Find the General Solution:**

The characteristic equation for the differential equation is:

$$
   r^2 - 1 = 0 \implies r = \pm 1
   $$

Therefore, the general solution is:

$$
   y(t) = A e^{t} + B e^{-t}
   $$

2. **Apply the Boundary Conditions:**

- **First Boundary Condition ($$ y(0) = 1 $$):**

Substitute $$ t = 0 $$ into the general solution:

$$
     y(0) = A e^{0} + B e^{0} = A + B = 1 \quad \Rightarrow \quad A + B = 1 \quad \text{(1)}
     $$

- **Second Boundary Condition ($$ y(\infty) = 0 $$):**

Examine the behavior of the solution as $$ t \to \infty $$:

$$
     y(t) = A e^{t} + B e^{-t}
     $$

For $$ y(t) $$ to approach 0 as $$ t \to \infty $$, the term $$ A e^{t} $$ must not diverge to infinity. This requires $$ A = 0 $$.

Substituting $$ A = 0 $$ into equation (1):

$$
     0 + B = 1 \implies B = 1
     $$

3. **Final Solution:**

Substituting $$ A = 0 $$ and $$ B = 1 $$ back into the general solution:

$$
   y(t) = e^{-t}
   $$

**Answer:** Its solution is the exponential function y (t ) = e⁻ᵗ
The solution to the differential equation is $$ y(t) = e^{-t} $$.

The solution of the differential equation

subject to the boundary conditions and , is

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The solution of the differential equation subject to the boundary conditions and , is