Question

dx/dy− x/​ y1=−xy³. Make a substitution to create a new DE which is linear and write it in standard form. Substitution: New DE:

Answer 1

To simplify the original differential equation dx/dy - x/y = -xy^3, the substitution u = x/y is made. This transforms the equation into a standard form linear differential equation du/dy + u · y^3 = 0.

Step-by-step explanation:

The original differential equation provided is dx/dy - x/y = -xy^3. To make a substitution that turns this into a linear differential equation, we can let u = x/y. With this substitution, dx/dy = du/dy · y + u since by the product rule, dx/dy = d(u · y)/dy = y · du/dy + u · dy/dy = y · du/dy + u. Substituting back into the original equation we get:

y · du/dy + u - u = -u · y^4

Which simplifies to:

y · du/dy = -u · y^4

Separating variables:

du/dy = -u · y^3

This forms a linear differential equation in the standard form:

du/dy + u · y^3 = 0