The general solution y(x) of the differential | vedclass

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Question

(B) Given the differential equation $\frac{d}{dy} \left( \int_x^y dt \right) = x$. Using the Fundamental Theorem of Calculus,$\frac{d}{dy} \int_x^y dt

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$y = -\ln|1 - x| + C$

Vedclass · Answer

(B) Given the differential equation $\frac{d}{dy} \left( \int_x^y dt \right) = x$. Using the Fundamental Theorem of Calculus,$\frac{d}{dy} \int_x^y dt = 1 - \frac{dx}{dy}$. Substituting this into the equation: $1 - \frac{dx}{dy} = x$. Rearranging the terms: $\frac{dx}{dy} = 1 - x$. Separating the variables: $\frac{dx}{1 - x} = dy$. Integrating both sides: $\int \frac{dx}{1 - x} = \int dy$. $-\ln|1 - x| = y + C_1$,which can be written as $y = -\ln|1 - x| + C$.

The general solution $y(x)$ of the differential equation $\frac{d}{dy} \left( \int_x^y dt \right) = x$ is:

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