Let y = y_1(x) and y = y_2(x) be the solution | vedclass

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(B) The given differential equation is $\frac{dy}{dx} - y = 7$. This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P =

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no point

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(B) The given differential equation is $\frac{dy}{dx} - y = 7$. This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = -1$ and $Q = 7$. The integrating factor is $I.F. = e^{\int -1 dx} = e^{-x}$. The general solution is $y \cdot e^{-x} = \int 7 e^{-x} dx + C = -7e^{-x} + C$. Thus,$y = Ce^x - 7$. For $y_1(0) = 0$: $0 = C_1(1) - 7 \Rightarrow C_1 = 7$. So,$y_1(x) = 7e^x - 7$. For $y_2(0) = 1$: $1 = C_2(1) - 7 \Rightarrow C_2 = 8$. So,$y_2(x) = 8e^x - 7$. To find the intersection,set $y_1(x) = y_2(x)$: $7e^x - 7 = 8e^x - 7$. $7e^x = 8e^x \Rightarrow e^x = 0$. Since $e^x$ is never $0$ for any real $x$,there is no point of intersection.

Let $y = y_1(x)$ and $y = y_2(x)$ be the solution curves of the differential equation $\frac{dy}{dx} = y + 7$ with initial conditions $y_1(0) = 0$ and $y_2(0) = 1$ respectively. Then the curves $y = y_1(x)$ and $y = y_2(x)$ intersect at

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