Find the equation of the curve for which y^('')=(4)/(x^(3)) and which is tangent to the line 2x+y=5 at the point (1,3)

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Find the equation of the curve for which  y^('')=(4)/(x^(3)) and which is tangent to the line  2x+y=5 at the point  (1,3)

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Integrate second derivative: : Integrate the second derivative to find the first derivative. We are given $ y'' = \frac{4}{x^3} $. To find the first derivative $ y' $, we integrate $ y'' $ with respect to $ x $. $$ y' = \int y'' \, dx = \int \frac{4}{x^3} \, dx = -\frac{2}{x^2} + C_1 $$Determine constant using tangent: : Determine the constant $ C_1 $ using the tangent condition. The curve is tangent to the line $ 2x + y = 5 $ at the point $ (1,3) $. The slope of the tangent line at this point is the derivative of $ y $ with respect to $ x $ at $ x = 1 $. The slope of the line $ 2x + y = 5 $ is $ -2 $ (since $ y = -2x + 5 $). So, $ y'(1) = -2 $. Substitute $ x = 1 $ into $ y' = -\frac{2}{x^2} + C_1 $ to find $ C_1 $. $$ -2 = -\frac{2}{1^2} + C_1 $$ $$ C_1 = -2 + 2 = 0 $$Integrate first derivative: : Integrate the first derivative to find the original function $ y $. Now that we know $ C_1 = 0 $, we have $ y' = -\frac{2}{x^2} $. To find $ y $, we integrate $ y' $ with respect to $ x $. $$ y = \int y' \, dx = \int -\frac{2}{x^2} \, dx = \frac{2}{x} + C_2 $$Determine constant using point: : Determine the constant $ C_2 $ using the point $ (1,3) $. The curve passes through the point $ (1,3) $. Substitute $ x = 1 $ and $ y = 3 $ into $ y = \frac{2}{x} + C_2 $ to find $ C_2 $. $$ 3 = \frac{2}{1} + C_2 $$ $$ C_2 = 3 - 2 = 1 $$Write final equation: : Write the final equation of the curve. Now that we have both constants $ C_1 $ and $ C_2 $, we can write the final equation of the curve. $$ y = \frac{2}{x} + 1 $$

$16$ . Find the equation of the curve for which $y^{\prime \prime}=\frac{4}{x^{3}}$ and which is tangent to the line $2 x+y=5$ at the point $(1,3)$

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161616. Find the equation of the curve for which y′′=4x3 y^{\prime \prime}=\frac{4}{x^{3}} y′′=x34​ and which is tangent to the line 2x+y=5 2 x+y=5 2x+y=5 at the point (1,3) (1,3) (1,3)

$16$ . Find the equation of the curve for which $y^{\prime \prime}=\frac{4}{x^{3}}$ and which is tangent to the line $2 x+y=5$ at the point $(1,3)$