Solving the Differential Equation y3y 18xy2 8xy2

Given the differential equation:

yy318xy2 8xy2

To simplify the equation, let's first rearrange it and solidify our understanding:

yy318xyy2 8xyy2

Note that:

yy318xy3 8xy3

Seperating the Equation

The next step is to separate the variables:

yy3 26xy3

Dividing both sides by y3v3 and x:

1yy2yy3 26xy2

Which simplifies to:

1y?2yy 18x?8x

Dividing both sides by 10:

1y?2yy 18x?8x

Integration Steps

Integrating Both Sides

We now integrate both sides of the equation:

∫xx226xy2y?2y dx ∫y18y 8x1y y?2y dy

Replacing y with tanθ

To integrate the left-hand side, we use the trigonometric substitution y tanθ, where dy sec2θ dθ:

∫θθ 1tan3θ sec2θtan2θ dθ

This simplifies to:

∫θθ 1tan3θ dθ

This can be further simplified to:

∫θθ 1tanθ tan2θ dθ ? tanθ dθ

Integrating

Integrating both terms:

∫θθ 1tanθ sec2θ dθ ? tanθ dθ

Yielding:

12tan2θ ? ln |cosθ| C

Substituting Back

Substituting back y tanθ, we find:

12y2 ? ln |cos1/2y| C

Simplifying further, the final solution becomes:

y2 ? e?2x?8x2 y2C

Conclusion

The detailed solution involves understanding the separable ODE and utilizing integration techniques, including trigonometric substitution. This method ensures that the given differential equation can be solved effectively, leading to the final integrated result.

Keywords: differential equation, separable ODE, exponential function