The substitution x=vy converts which one of t | vedclass

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(D) The substitution $x=vy$ is used to solve homogeneous differential equations of the form $\frac{dx}{dy} = f(\frac{x}{y})$ or $\frac{dy}{dx} = g(\fr

Vedclass · Accepted Answer

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

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(D) The substitution $x=vy$ is used to solve homogeneous differential equations of the form $\frac{dx}{dy} = f(\frac{x}{y})$ or $\frac{dy}{dx} = g(\frac{x}{y})$.For the substitution $x=vy$ (which implies $\frac{x}{y}=v$),the differential equation must be homogeneous in $x$ and $y$ such that $\frac{dy}{dx}$ can be expressed as a function of $\frac{x}{y}$.Checking option $(d)$:$(1+2e^{\frac{x}{y}}) + 2e^{\frac{x}{y}}(1-\frac{x}{y})\frac{dy}{dx} = 0$$\frac{dy}{dx} = -\frac{1+2e^{\frac{x}{y}}}{2e^{\frac{x}{y}}(1-\frac{x}{y})}$Since the right-hand side is a function of $\frac{x}{y}$,it is a homogeneous differential equation of degree zero,which becomes separable after substituting $x=vy$.

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

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