Solve the equation. StartFraction dy Over dx EndFraction equals5 x Superscript 4 Baseline (1 plus y squared) Superscript three halves An implicit solution in the form F (x, y) equalsC is nothingequals C, where C is an arbitrary constant.

Step-by-step explanation:

To solve the differential equation

dy/dx = 5x^4 (1 + y²) ^ (3/2)

First, separate the variables

dy / (1 + y²) ^ (3/2) = 5x^4 dx

Now, integrate both sides

To integrate dy / (1 + y²) ^ (3/2), use the substitution y = tan (u)

dy = (1/cos²u) du

So,

dy / (1 + y²) ^ (3/2) = [ (1/cos²u) / (1 + tan²u) ^ (3/2) ]du

= (1/cos²u) / (1 + (sin²u/cos²u)) ^ (3/2)

Because cos²u + sin²u = 1 (Trigonometric identity),

The equation becomes

[1 / (1/cos²u) ^ (3/2) * 1/cos²u] du

= cos³u/cos²u

= cosu

Integral of cosu = sinu

But y = tanu

Therefore u = arctany

We then have

cos (arctany) = y/√ (1 + y²)

Now, the integral of the equation

dy / (1 + y²) ^ (3/2) = 5x^4 dx

Is

y/√ (1 + y²) = x^5 + C

So

y - (x^5 + C) √ (1 + y²) = 0

is the required implicit solution