The differential equation
has an implicit general solution of the form , where is an arbitary constant.
In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form

Find such a solution and then give the related functions requested.

Ask by Gardner Henry. in the United States

Mar 21,2025

Question

The differential equation
has an implicit general solution of the form , where is an arbitary constant.
In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form

Find such a solution and then give the related functions requested.

Ask by Gardner Henry. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

To find the implicit general solution of a separable differential equation in the form $$ F(x, y) = G(x) + H(y) = K $$, we need to follow these steps:

1. **Identify the differential equation**: Since the specific differential equation is not provided, I will assume a general separable form $$ \frac{dy}{dx} = f(x)g(y) $$.

2. **Separate the variables**: Rearranging gives us $$ \frac{1}{g(y)} dy = f(x) dx $$.

3. **Integrate both sides**: We will integrate the left side with respect to $$ y $$ and the right side with respect to $$ x $$.

4. **Define the functions $$ G(x) $$ and $$ H(y) $$**: After integration, we will express the solution in the form $$ F(x, y) = G(x) + H(y) = K $$.

Since the specific form of the differential equation is not given, I will proceed with a general example. Let's assume the differential equation is:

$$
\frac{dy}{dx} = xy
$$

### Step 1: Separate the variables
$$
\frac{1}{y} dy = x dx
$$

### Step 2: Integrate both sides
Integrating both sides gives:
$$
\int \frac{1}{y} dy = \int x dx
$$

This results in:
$$
\ln |y| = \frac{x^2}{2} + C
$$

### Step 3: Solve for $$ y $$
Exponentiating both sides, we have:
$$
|y| = e^{\frac{x^2}{2} + C} = e^C e^{\frac{x^2}{2}}
$$
Let $$ K = e^C $$, then:
$$
y = K e^{\frac{x^2}{2}}
$$

### Step 4: Define $$ G(x) $$ and $$ H(y) $$
We can express this in the form $$ F(x, y) = G(x) + H(y) = K $$:
- Let $$ G(x) = \frac{x^2}{2} $$
- Let $$ H(y) = \ln |y| $$

Thus, the implicit solution can be written as:
$$
F(x, y) = \frac{x^2}{2} + \ln |y| = K
$$

If you have a specific differential equation in mind, please provide it, and I can give a more tailored solution.
$$ F(x, y) = \frac{x^2}{2} + \ln |y| = K $$

The differential equation
has an implicit general solution of the form , where is an arbitary constant.
In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form

Find such a solution and then give the related functions requested.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1: Separate the variables

Step 2: Integrate both sides

Step 3: Solve for

Step 4: Define and

Beyond the Answer

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Latest Calculus Questions

The differential equation has an implicit general solution of the form , where is an arbitary constant. In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form Find such a solution and then give the related functions requested.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1: Separate the variables

Step 2: Integrate both sides

Step 3: Solve for

Step 4: Define and

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions

The differential equation
has an implicit general solution of the form , where is an arbitary constant.
In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form

Find such a solution and then give the related functions requested.