Let y'(x) + y(x) g'(x) = g(x) g'(x), y (0) = 0, x ∈ R, where f '(x) denotes \frac{d \textrm{ } f \left(\right. x \left.\right)}{d \left(\right. x \left.\right)} and g (x) is a given non‑constant differentiable function on R with g (0) = g (2) = 0. Then the value of y(2) is

Engineering

Mathematics

Linear Differential Equation

Question

Let y'(x) + y(x) g'(x) = g(x) g'(x), y (0) = 0, x ∈ R, where f '(x) denotes $\frac{d f (x)}{d (x)}$ and g (x) is a given non‑constant differentiable function on R with g (0) = g (2) = 0. Then the value of y(2) is

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$\frac{dy}{dg} + y = g$

I. F. = $\int_{}^{}$ 1.dg = g

y.e^g = $\int_{}^{}$ ge^g .de^g – $\int$ e^g.d^g

ye^g = ge^g – e^g + c

y = g –1 + ce^–g

$∵$ y(0) = 0 & g(0) = 0

at x = 0

0 = 0 – 1 + Ce^–0

C = 1

y = g – 1 + e^–g

at x = 2

y(2) = 0 – 1 + e^–0 = 0