Let y = y 1 ( x ) and y = y 2 ( x ) be the solution curves of the differential equation d y d x = 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

Question

Let $y = y_{1} (x)$ and $y = y_{2} (x)$ be the solution curves of the differential equation $\frac{d y}{d x} = y + 7$ 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 [2023]

Smart Achievers · Accepted Answer

(3)

We have, $\frac{d y}{d x} = y + 7, y_{1} (0) = 0, y_{2} (0) = 1$

$\int \frac{d y}{y + 7} = \int d x$

$\ln$ $| y + 7 |$ $= x + k \Rightarrow y + 7 = e^{x + k} = e^{x} \cdot e^{k} = C e^{x} [∵ e^{k} = C]$

$\Rightarrow y = - 7 + C e^{x}$

$y_{1} (0) = 0 \Rightarrow 0 = - 7 + C \Rightarrow C = 7$

$y_{2} (0) = 1 \Rightarrow 1 = - 7 + C \Rightarrow C = 8$

$∴$ $y_{1} (x) = - 7 + 7 e^{x} and y_{2} (x) = - 7 + 8 e^{x}$

If $y_{1} (x)$ and $y_{2} (x)$ intersect at any point, then the values of both curves will be the same at that point.

$∴$ $- 7 + 7 e^{x} = - 7 + 8 e^{x} \Rightarrow e^{x} = 0 \Rightarrow Not possible$

$∴$ $The curves y_{1} (x) and y_{2} (x) will not intersect at any point.$

Solution

Q.
Let $y = y_{1} (x)$ and $y = y_{2} (x)$ be the solution curves of the differential equation $\frac{d y}{d x} = y + 7$ 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 [2023]

1 infinite number of points

2 two points

3 no point

4 one point

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