For x∈ℝ, let y(x) be a solution of the differential equation x2-5dydx-2xy=-2xx2-52 such that y(2)=7Then the maximum value of the function y(x) is

x2−5dydx−2xy=−2xx2−52y2=7yx2−5=−∫2xdxyx2−5=−x2+c74−5=−4+C⇒C=−7+4C=−3+Y=15+2X2−X416−X2−12ymax=16

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

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answer is 16.

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Detailed Solution

$\begin{array}{l} (x^{2} - 5) \frac{dy}{dx} - 2 xy = - 2 x {(x^{2} - 5)}^{2} \\ y (2) = 7 \\ (\frac{y}{x^{2} - 5}) = - \int 2 xdx \\ \frac{y}{x^{2} - 5} = - x^{2} + c \\ \frac{7}{4 - 5} = - 4 + C \Rightarrow C = - 7 + 4 \\ C = - 3 + Y = 15 + 2 X^{2} - X^{4} \\ 16 - {[X^{2} - 1]}^{2} \\ y_{\max} = 16 \end{array}$