How do you use the differential equation #dy/dx=48/(3x+5)^3# to find the equation of the function given point (1,3)?
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To use the given differential equation ( \frac{dy}{dx} = \frac{48}{(3x + 5)^3} ) to find the equation of the function given the point ( (1,3) ), follow these steps:
 Integrate both sides of the differential equation with respect to ( x ) to find the general solution.
 Use the given point ( (1,3) ) to find the particular solution by substituting the coordinates into the general solution and solving for the constant of integration.
Here are the steps in detail:
 Integrate both sides of the differential equation: [ \int \frac{dy}{dx} , dx = \int \frac{48}{(3x + 5)^3} , dx ] [ y = \int \frac{48}{(3x + 5)^3} , dx ]
To integrate ( \frac{48}{(3x + 5)^3} ) with respect to ( x ), perform a substitution. Let ( u = 3x + 5 ), then ( du = 3dx ). Rewrite the integral in terms of ( u ): [ y = \frac{48}{3} \int \frac{1}{u^3} , du ] [ y = 16 \int u^{3} , du ] [ y = 16 \left( \frac{u^{2}}{2} \right) + C ] [ y = 8u^{2} + C ]
Now, substitute back ( u = 3x + 5 ): [ y = 8(3x + 5)^{2} + C ]
 Use the given point ( (1,3) ) to find the constant of integration ( C ): [ 3 = 8(3(1) + 5)^{2} + C ] [ 3 = 8(3(1) + 5)^{2} + C ] [ 3 = 8(8)^{2} + C ] [ 3 = 8 \left( \frac{1}{64} \right) + C ] [ 3 = \frac{1}{8} + C ] [ C = 3  \frac{1}{8} ] [ C = \frac{23}{8} ]
Therefore, the particular solution of the differential equation is: [ y = 8(3x + 5)^{2} + \frac{23}{8} ]
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To find the equation of the function given the point (1,3) and the differential equation dy/dx = 48/(3x + 5)^3, follow these steps:

Integrate the given differential equation with respect to x to find the function y(x). ∫ dy = ∫ 48/(3x + 5)^3 dx

After integrating, you'll obtain an equation involving y and x.

Next, apply the initial condition (the given point (1,3)) to determine the constant of integration.

Substitute the x and y values of the given point into the equation obtained from integration to solve for the constant.

Once you have the constant, substitute it back into the equation obtained from integration to get the specific equation of the function.
Following these steps will yield the equation of the function given the point (1,3) and the provided differential equation.
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