Solving a Differential Equation | Rewriting and Integrating Steps

y’=(-14x)/(x²+9)²

To find a solution to this differential equation, we must first rewrite it in a form that is easier to work with

To find a solution to this differential equation, we must first rewrite it in a form that is easier to work with. Let’s break it down step by step.

First, notice that the equation is given in what is called “differential notation,” where y’ denotes the derivative of y with respect to x. To make it more familiar, let’s rewrite the equation using Leibniz notation, where dy/dx denotes the derivative of y with respect to x:

dy/dx = (-14x)/(x²+9)²

Now, to proceed with solving the differential equation, we will separate variables and integrate both sides.

First, multiply both sides of the equation by dx to move the differential term:
dy = (-14x)/(x²+9)² dx

Now, let’s integrate both sides:
∫dy = ∫(-14x)/(x²+9)² dx

On the left side, integrating ∫dy with respect to y simply gives us y:
y = ∫(-14x)/(x²+9)² dx

The right side is now an integral of a rational function. To solve it, we can use substitution. Let u = x² + 9, which implies du = 2x dx. Rearranging this equation, x dx = (1/2) du.

Substituting into the integral, we have:
y = ∫(-14/2) du/u²
y = -7 ∫ du/u²

Integrating ∫du/u² gives us -1/u:
y = -7(-1/u) + C

Replacing u with x² + 9:
y = 7/(x² + 9) + C

Therefore, the general solution to the given differential equation is:
y = 7/(x² + 9) + C, where C is an arbitrary constant.

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