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Question

Mathematics Question on Differential equations

For xRx ∈ R, let y(x)y(x) be the solution of the differential equation (x^2 -5)$$\frac{dy}{dx}$$-2xy = - 2x$$(x^2-5)^2 such that y(2)=7y(2) =7. Then the maximum value of the function y(x)y(x) is

Answer

Given:
(x^2 -5)$$\frac{dy}{dx}-2xy = - 2x(x25)2(x^2-5)^2
Rearrange the equation to solve for dydx\frac{dy}{dx}​:
dydx=2x(x25)22xyx25\frac{dy}{dx}=\frac{2x(x^2-5)^2-2xy}{x^2-5}
Now, we have a first-order separable differential equation. Let's separate variables:
dy2x(x25)22xy=dxx25\frac{dy}{{2x(x^2-5)^2-2xy}}=\frac{dx}{x^2-5}
Next, integrate both sides:
dy2x(x25)22xy=dxx25\int\frac{dy}{{2x(x^2-5)^2-2xy}}=\int\frac{dx}{x^2-5}
Integrating these expressions will provide us with the function y(x).