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PROBLEM 1

dy/dx = y(x^3 - √x)

I have separated the variables as follows:

Rewrote equation as dy/dx = y(x^3 - x^1/2)

Divided both sides by 1/y

1/y dy/dx = x^3 - x^1/2

Multiplied both sides by dx

1/y dy = x^3 - x^1/2 dx

Then to find the indefinite integral

∫1/y dy = ∫x^3 - x^1/2 dx

ln[y] = 0.25x^4 - 1.5x^3/2

y = e^(0.25x^4 - 1.5x^3/2) + C

Does this look correct?

PROBLEM 2

(yx^4 - yx^2) = dy/dx y^2.x^4

Separating variables:

Multiply both sides by dx

(yx^4 - yx^2) dx = y^2.x^4 dy

Divide both sides by y

x^4 - x^2 dx = y.x^4 dy

Divide boths sides by x^4

(x^4 - x^2)/x^4 dx = y dy

Swap LHS and RHS so

y dy = (x^4 - x^2)/x^4 dx

Then to find the indefinite integral

∫y dy = ∫(x^4 - x^2)/x^4 dx

y^2/2 = (x^2 + 1)/x + C

Solve for y

y = √2(x^2 + 1)/x + C

If my solutions are incorrect then any pointers in the right direction would be gratefully received.

Thanks