The substitution x=vy converts which one of the following differential equation to an equation solvable by variable separable method?

1 + 2e^x/y + 2e^x/y 1 − x/y dy/dx = 0

To determine which equation can be solved by the variable separable method using the substitution x = vy, we need to analyze the given options for homogeneity. The substitution x = vy is particularly effective for transforming homogeneous differential equations into separable ones.

A differential equation is homogeneous if all its terms can be expressed in terms of a single power of y/x or x/y. This means that substituting x = vy simplifies the equation, allowing variables to be separated.

Let's analyze the options:

Option (a): y² − 2x²y dx = x² − 2xy² dy

Not homogeneous. The terms do not exhibit proportionality when expressed as a function of y/x.

Option (b): x² dy − y dx = x² + y² dx

Not homogeneous. The presence of x² + y² disrupts homogeneity.

Option (c): dy/dx = y²/x + xy

Not homogeneous. The term y²/x indicates a non-homogeneous structure.

Option (d): (1 + 2e^xy) + 2e^xy(1 − xy) (dy/dx) = 0

This is a homogeneous differential equation. Substituting x = vy will transform this equation into a separable form, making it solvable by the variable separable method.

Only Option D is a homogeneous differential equation that can be converted into a separable form using the substitution x = vy.