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Step-by-Step Solution
Step 1: Write down the given information
We have a curve passing through the origin, i.e., it satisfies:
When $x = 0$, then $y = 0.$
The slope of the tangent (the derivative $dy/dx$) at any point $(x,y)$ on the curve is:
$ \frac{dy}{dx} = \frac{x^2 - 4x + y + 8}{x - 2}. $
Step 2: Rewrite and rearrange the differential equation
Rearrange the equation so that $y$-terms appear together:
$ \frac{dy}{dx} = \frac{x^2 - 4x + y + 8}{x - 2}. $
Subtract the term involving $y$ on the right-hand side:
$ \frac{dy}{dx} - \frac{y}{x - 2} = \frac{x^2 - 4x + y + 8}{x - 2} - \frac{y}{x - 2}
= \frac{x^2 - 4x + 8}{x - 2}. $
Notice that
$ x^2 - 4x + 8 = (x - 2)^2 + 4. $
Dividing by $(x - 2)$, we get
$ \frac{(x - 2)^2 + 4}{x - 2} = (x - 2) + \frac{4}{x - 2}. $
Hence, the differential equation in standard linear form is:
$ \frac{dy}{dx} - \frac{y}{x - 2} = (x - 2) + \frac{4}{x - 2}. $
Step 3: Find the integrating factor (I.F.)
For a linear differential equation of the form
$ \frac{dy}{dx} + P(x)\,y = Q(x), $
the integrating factor is
$ e^{\int P(x)\,dx}. $
Here, $P(x) = -\frac{1}{x - 2}$. Thus,
$ \int P(x)\,dx = \int \left(-\frac{1}{x - 2}\right)\,dx = -\ln|x - 2|. $
Therefore, the integrating factor (I.F.) is:
$ e^{-\ln|x - 2|} = \frac{1}{x - 2}. $
Step 4: Multiply the differential equation by the integrating factor
Multiply both sides of
$ \frac{dy}{dx} - \frac{y}{x - 2} = (x - 2) + \frac{4}{x - 2} $
by $ \frac{1}{x - 2} $:
$ \frac{1}{x - 2}\,\frac{dy}{dx} - \frac{y}{(x - 2)^2}
= \frac{1}{x - 2}\left[(x - 2) + \frac{4}{x - 2}\right]. $
The left-hand side becomes the derivative of
$ \frac{y}{x - 2} $ with respect to $x$:
$ \frac{d}{dx}\Bigl(\frac{y}{x - 2}\Bigr). $
Thus,
$ \frac{d}{dx}\Bigl(\frac{y}{x - 2}\Bigr)
= \frac{1}{x - 2}\left[(x - 2) + \frac{4}{x - 2}\right]. $
Step 5: Integrate both sides
Now integrate both sides with respect to $x$:
$ \frac{y}{x - 2} = \int \left[1 + \frac{4}{(x - 2)^2}\right] \,dx. $
The integral of the right-hand side:
$ \int 1 \, dx = x, $
and
$ \int \frac{4}{(x - 2)^2}\,dx = 4 \int (x - 2)^{-2}\,dx
= 4\left(-\frac{1}{x - 2}\right) = -\frac{4}{x - 2}. $
Hence,
$ \frac{y}{x - 2} = x - \frac{4}{x - 2} + C, $
where $C$ is the constant of integration.
Step 6: Substitute the initial condition to find C
The curve passes through the origin $(0,0)$. Substitute $x=0$, $y=0$:
$ \frac{0}{0 - 2} = 0 - \frac{4}{0 - 2} + C. $
This simplifies to
$ 0 = - \left(-2\right) + C = 2 + C. $
Therefore, $ C = -2. $
Step 7: Write the final solution for y
Substituting $C = -2$ back:
$ \frac{y}{x - 2} = x - \frac{4}{x - 2} - 2. $
Multiply both sides by $(x - 2)$:
$ y = (x - 2)\,\bigl(x - \frac{4}{x - 2} - 2\bigr). $
Simplify carefully:
$ y = (x - 2)\,x \;-\; (x - 2)\,\frac{4}{x - 2} \;-\; 2(x - 2). $
$ y = x(x - 2) - 4 - 2(x - 2). $
$ y = x^2 - 2x - 4 - 2x + 4. $
$ y = x^2 - 4x. $
Step 8: Verify the additional point (5, 5)
To check if the curve passes through $(5,5)$, substitute $x = 5$:
$ y = 5^2 - 4\cdot 5 = 25 - 20 = 5. $
Therefore, the point $(5,5)$ lies on the curve.
Conclusion
The curve that satisfies the given slope condition and passes through the origin also passes through the point $(5,5).$
