(A) Given the differential equation: $\frac{dt}{dx} = \frac{t}{x + t e^{-2x/t}}$.Taking the reciprocal of both sides,we get:$\frac{dx}{dt} = \frac{x +

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(A) Given the differential equation: $\frac{dt}{dx} = \frac{t}{x + t e^{-2x/t}}$.Taking the reciprocal of both sides,we get:$\frac{dx}{dt} = \frac{x + t e^{-2x/t}}{t}$.Dividing each term in the numerator by $t$,we obtain:$\frac{dx}{dt} = \frac{x}{t} + \frac{t e^{-2x/t}}{t}$.Simplifying the expression,we get:$\frac{dx}{dt} = \frac{x}{t} + e^{-2(x/t)}$.This is in the form $\frac{dx}{dt} = \phi\left(\frac{x}{t}\right)$,where $\phi(v) = v + e^{-2v}$.Thus,the correct option is $A$.

Class 12 Mathematics Differential Equations Homogeneous differential equations

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Express $\frac{dt}{dx} = \frac{t}{x + t e^{-2x/t}}$ in the form of $\frac{dx}{dt} = \phi\left(\frac{x}{t}\right)$.

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A
$\frac{x}{t} + e^{-2(x/t)}$
B
$\frac{x}{t} - e^{-2(x/t)}$
C
$\frac{x}{t} + e^{2(x/t)}$
D
$\frac{x}{t} - e^{2(x/t)}$

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Express $\frac{dt}{dx} = \frac{t}{x + t e^{-2x/t}}$ in the form of $\frac{dx}{dt} = \phi\left(\frac{x}{t}\right)$.

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