Solve the equation. $$ \frac{d y}{d x}=-\frac{e^{x} y^{2}}{10} $$

Question

UpStudy AI · Accepted Answer

To solve the differential equation:

$$
\frac{dy}{dx} = -\frac{e^{x} y^{2}}{10}
$$

we can follow these steps:

### Step 1: Separate Variables
The equation is separable, meaning we can rearrange it to isolate $$ y $$ terms on one side and $$ x $$ terms on the other:

$$
\frac{dy}{y^{2}} = -\frac{e^{x}}{10} dx
$$

### Step 2: Integrate Both Sides
Integrate both sides of the equation:

$$
\int \frac{dy}{y^{2}} = \int -\frac{e^{x}}{10} dx
$$

**Left Side:**
$$
\int y^{-2} dy = -y^{-1} + C_1 = -\frac{1}{y} + C_1
$$

**Right Side:**
$$
\int -\frac{e^{x}}{10} dx = -\frac{e^{x}}{10} + C_2
$$

### Step 3: Combine Constants
Combine the constants of integration into a single constant $$ C $$:

$$
-\frac{1}{y} = -\frac{e^{x}}{10} + C
$$

Multiply both sides by $$-1$$ to simplify:

$$
\frac{1}{y} = \frac{e^{x}}{10} - C
$$

Let's redefine the constant as $$ C' = -C $$ for simplicity:

$$
\frac{1}{y} = \frac{e^{x}}{10} + C'
$$

### Step 4: Solve for $$ y $$
Finally, solve for $$ y $$:

$$
y = \frac{1}{\frac{e^{x}}{10} + C'} = \frac{10}{e^{x} + 10C'}
$$

Since $$ C' $$ is an arbitrary constant, we can simplify the expression by renaming it as $$ C $$:

$$
y = \frac{10}{e^{x} + C}
$$

### Final Answer

After simplifying, a general solution is  y = 10 divided by (eˣ plus a constant). Thus,

y = 10 ⁄ (e^x + C)
The solution to the differential equation is $$ y = \frac{10}{e^{x} + C} $$, where $$ C $$ is a constant.

Solve the equation.

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