JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Differential Equations

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

y_{2} (0) = 1

respectively. Then the curves

y = y_{1} (x)

and

y = y_{2} (x)

intersect at

Answer: (B)

Solution

The differential equation

\frac{d y}{d x} - y = 7

is linear. The integrating factor is

e^{- x}

. Multiplying:

y e^{- x} = \int 7 e^{- x} d x = - 7 e^{- x} + C

, so

y = - 7 + C e^{x}

. For

y_{1} (0) = 0

0 = - 7 + C

, giving

C = 7

, so

y_{1} = - 7 + 7 e^{x}

. For

y_{2} (0) = 1

1 = - 7 + C

, giving

C = 8

, so

y_{2} = - 7 + 8 e^{x}

. Setting

y_{1} = y_{2}

7 e^{x} = 8 e^{x}

, which gives

e^{x} = 0

, impossible. Thus the curves do not intersect.

JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Single correct option

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