If for $x \geq 0$, $y=y(x)$ is the solution of the differential equation, $(x+1) d y=\left((x+1)^{2}+y-3\right) d x$, $y(2)=0$, then $y(3)$ is equal to ______

JEE Main · 2020 · 09 Jan 2020 · Shift 1 · Differential Equations

If for

x \geq 0

y = y (x)

is the solution of the differential equation,

(x + 1) d y = ((x + 1)^{2} + y - 3) d x

y (2) = 0

, then

y (3)

is equal to ______

Answer: 3

Rearranged, the equation is linear:

\frac{d y}{d x} - \frac{y}{x + 1} = (x + 1) - \frac{3}{x + 1}

, with integrating factor

\frac{1}{x + 1}

. It becomes

\frac{d}{d x} (\frac{y}{x + 1}) = 1 - \frac{3}{( x + 1 ) ^{2}}

, so

\frac{y}{x + 1} = x + \frac{3}{x + 1} + C

and

y = x (x + 1) + 3 + C (x + 1)

. The condition

y (2) = 0

gives

9 + 3 C = 0

, so

C = - 3

and

y = x^{2} - 2 x

. Hence

y (3) = 9 - 6 = 3

JEE Main · 2020 · 09 Jan 2020 · Shift 1 · Integer answer

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