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

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$(y^{2} - 2 x^{2} y) dx = (x^{2} - 2 {xy}^{2}) dy$

$\frac{dy}{dx} = \frac{y^{2}}{x + \sqrt{xy}}$

$x^{2} dy - ydx = \sqrt{x^{2} + y^{2}} dx$

$(1 + 2 e^{\frac{x}{y}}) + 2 e^{\frac{x}{y}} (1 - \frac{x}{y}) \frac{dy}{dx} = 0$

answer is D.

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Detailed Solution

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.

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