Find the implicit general solution to the differential equation y^5 (dy/dx) = (y^6 + 2) cos(x) An implicit general solution in the form F(x,y) = C is _____=C , where C is an arbitratary constant

To find the implicit general solution to the given differential equation, we follow these steps:

### Given Differential Equation:
\[
y^5 \frac{dy}{dx} = (y^6 + 2) \cos(x)
\]

### Step 1: Separate the Variables
We want to separate the variables \( y \) and \( x \). We can rewrite the equation as:

\[
\frac{y^5 \, dy}{y^6 + 2} = \cos(x) \, dx
\]

### Step 2: Integrate Both Sides
Now, integrate both sides with respect to their respective variables.

For the left side:

\[
\int \frac{y^5 \, dy}{y^6 + 2}
\]

Let \( u = y^6 + 2 \), so \( du = 6y^5 \, dy \).

The integral then becomes:

\[
\frac{1}{6} \int \frac{du}{u} = \frac{1}{6} \ln|u| + C_1 = \frac{1}{6} \ln|y^6 + 2| + C_1
\]

For the right side:

\[
\int \cos(x) \, dx = \sin(x) + C_2
\]

### Step 3: Combine the Results
Now, combining the integrals, we have:

\[
\frac{1}{6} \ln|y^6 + 2| = \sin(x) + C
\]

where \( C \) is the arbitrary constant that absorbs both \( C_1 \) and \( C_2 \).

### Step 4: Solve for the Implicit General Solution
To express the solution in the implicit form \( F(x, y) = C \), we multiply both sides by 6 to get rid of the coefficient:

\[
\ln|y^6 + 2| = 6\sin(x) + 6C
\]

We can simplify this by absorbing the constant into the right side:

\[
\ln|y^6 + 2| = 6\sin(x) + C'
\]

Exponentiating both sides to eliminate the logarithm, we obtain:

\[
|y^6 + 2| = e^{6\sin(x) + C'} = e^{6\sin(x)} \cdot e^{C'}
\]

Let \( C = e^{C'} \) (since \( C \) is still an arbitrary constant), the implicit general solution becomes:

\[
y^6 + 2 = C e^{6\sin(x)}
\]

### Final Answer:

The **implicit general solution** is:

\[
F(x, y) = y^6 + 2 = C e^{6\sin(x)}
\]

Would you like any more details or have any questions?

### Related Questions:

1. How do you perform variable substitution in integrals?
2. What is the significance of the arbitrary constant \( C \) in solutions to differential equations?
3. How do you solve a separable differential equation?
4. What is the relationship between implicit and explicit solutions?
5. Can you find the explicit solution for \( y \) in terms of \( x \) from this implicit solution?
6. How would the solution change if the differential equation was non-separable?
7. What are some common techniques for integrating functions involving trigonometric expressions?
8. How does the logarithmic function help in solving differential equations?

### Tip:
In implicit solutions, try to avoid unnecessary steps for simplifying the arbitrary constant \( C \), as it can absorb additive and multiplicative constants naturally.

Discover how to find the implicit general solution to the differential equation y^5 (dy/dx) = (y^6 + 2) cos(x). Follow a detailed step-by-step solution involving separating variables, integrating both sides, and simplifying to derive the implicit general solution. Suitable for advanced undergraduate students and those studying differential equations.

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