If $y(x)$ is the solution of the differential equation $$ x dy - (y^2 - 4y) dx = 0 \text{ for } x > 0, y(1) = 2, $$ and the slope of the curve $y = y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is ____.

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Differential Equations

y (x)

is the solution of the differential equation

x d y - (y^{2} - 4 y) d x = 0 for x > 0, y (1) = 2,

and the slope of the curve

y = y (x)

is never zero, then the value of

10 y (2)

is ____.

Answer: 8

Separating variables,

\frac{d y}{y ^{2} - 4 y} = \frac{d x}{x}

, and partial fractions give

\frac{1}{4} ln \frac{y - 4}{y} = ln x + C

. The condition

y (1) = 2

makes

C = 0

, so

\frac{y - 4}{y} = x^{4}

. Keeping the branch through

(1, 2)

\frac{y - 4}{y} = - x^{4}

, which gives

y = \frac{4}{1 + x ^{4}}

; this is positive for all

x > 0

, so the slope never vanishes. At

x = 2

y = \frac{4}{1 + 4} = \frac{4}{5}

, hence

10 y (2) = 8

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Integer answer