Find the solution of the initial value problem $$ \frac{d y}{d x}=(x-3) e^{-2 y}, \quad y(3)=\ln (3) $$ $$ y(x)= $$

Question

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To solve the initial value problem given by the differential equation

$$
\frac{d y}{d x} = (x - 3) e^{-2y}
$$

with the initial condition $$ y(3) = \ln(3) $$, we will follow these steps:

1. **Separate the variables**: Rearrange the equation to isolate $$ y $$ and $$ x $$.
2. **Integrate both sides**: Perform the integration.
3. **Apply the initial condition**: Use the initial condition to find the constant of integration.
4. **Solve for $$ y $$**: Express $$ y $$ in terms of $$ x $$.

### Step 1: Separate the variables

We can rewrite the equation as:

$$
e^{2y} dy = (x - 3) dx
$$

### Step 2: Integrate both sides

Now we integrate both sides:

$$
\int e^{2y} dy = \int (x - 3) dx
$$

Calculating the integrals:

- The left side: 
$$
\int e^{2y} dy = \frac{1}{2} e^{2y} + C_1
$$

- The right side:
$$
\int (x - 3) dx = \frac{1}{2} x^2 - 3x + C_2
$$

Setting both integrals equal gives us:

$$
\frac{1}{2} e^{2y} = \frac{1}{2} x^2 - 3x + C
$$

where $$ C = C_2 - C_1 $$.

### Step 3: Apply the initial condition

Now we apply the initial condition $$ y(3) = \ln(3) $$:

Substituting $$ x = 3 $$ and $$ y = \ln(3) $$:

$$
\frac{1}{2} e^{2 \ln(3)} = \frac{1}{2} (3^2) - 3(3) + C
$$

Calculating $$ e^{2 \ln(3)} = 3^2 = 9 $$:

$$
\frac{1}{2} \cdot 9 = \frac{1}{2} \cdot 9 - 9 + C
$$

This simplifies to:

$$
\frac{9}{2} = \frac{9}{2} - 9 + C
$$

Solving for $$ C $$:

$$
C = 9
$$

### Step 4: Solve for $$ y $$

Now we substitute $$ C $$ back into our equation:

$$
\frac{1}{2} e^{2y} = \frac{1}{2} x^2 - 3x + 9
$$

Multiplying through by 2:

$$
e^{2y} = x^2 - 6x + 18
$$

Taking the natural logarithm of both sides:

$$
2y = \ln(x^2 - 6x + 18)
$$

Finally, solving for $$ y $$:

$$
y = \frac{1}{2} \ln(x^2 - 6x + 18)
$$

Thus, the solution to the initial value problem is:

$$
y(x) = \frac{1}{2} \ln(x^2 - 6x + 18)
$$
$$ y(x) = \frac{1}{2} \ln(x^2 - 6x + 18) $$

Find the solution of the initial value problem

Upstudy AI Solution

Answer

Solution

Step 1: Separate the variables

Step 2: Integrate both sides

Step 3: Apply the initial condition

Step 4: Solve for

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