Let $\mathrm{y}=\mathrm{y}_{1}(\mathrm{x})$ and $\mathrm{y}=\mathrm{y}_{2}(\mathrm{x})$ be two distinct solutions of the differential equation $\frac{d y}{d x}=x+y$, with $\mathrm{y}_{1}(0)=0$ and $\mathrm{y}_{2}(0)=1$ respectively. Then, the number of points of intersection of $y=y_{1}(x)$ and $\mathrm{y}=\mathrm{y}_{2}(\mathrm{x})$ is

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Differential Equations

Let

y = y_{1} (x)

and

y = y_{2} (x)

be two distinct solutions of the differential equation

\frac{d y}{d x} = x + y

, with

y_{1} (0) = 0

and

y_{2} (0) = 1

respectively. Then, the number of points of intersection of

y = y_{1} (x)

and

y = y_{2} (x)

Answer: (A)

The differential equation

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

is linear with integrating factor

e^{- x}

.

General solution:

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

y = - x - 1 + c e^{x}

For

y_{1} (0) = 0

0 = - 1 + c

, so

c = 1

and

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

.

For

y_{2} (0) = 1

1 = - 1 + c

, so

c = 2

and

y_{2} = - x - 1 + 2 e^{x}

.

Now

y_{2} - y_{1} = e^{x} > 0

for all

x

, so

y_{2} > y_{1}

everywhere and they never intersect. Answer: (A)

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option

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