Key Concepts and Formulas

• First-Order Linear Differential Equation: A differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x)$ and $Q(x)$ are functions of $x$.
• Integrating Factor: For a first-order linear differential equation, the integrating factor is $e^{\int P(x) dx}$. Multiplying the differential equation by the integrating factor allows us to solve it.
• General Solution: After multiplying by the integrating factor, the general solution is given by $y \cdot (\text{Integrating Factor}) = \int Q(x) \cdot (\text{Integrating Factor}) dx + C$, where C is the constant of integration.

Step-by-Step Solution

Step 1: Rewrite the given differential equation.

We are given $\frac{dy}{dx} = \frac{x^2 - 4x + y + 8}{x - 2}$. Our goal is to rearrange it into a standard form that we can recognize and solve.

$\frac{dy}{dx} = \frac{x^2 - 4x + 4 + y + 4}{x - 2}$ $\frac{dy}{dx} = \frac{(x - 2)^2 + y + 4}{x - 2}$ $\frac{dy}{dx} = (x - 2) + \frac{y + 4}{x - 2}$ $\frac{dy}{dx} - \frac{1}{x - 2}y = x - 2 + \frac{4}{x-2}$

Step 2: Identify the type of differential equation and find the integrating factor.

The equation is a first-order linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x) = -\frac{1}{x - 2}$ and $Q(x) = x - 2 + \frac{4}{x-2}$.

The integrating factor (IF) is given by $e^{\int P(x) dx}$: $\text{IF} = e^{\int -\frac{1}{x - 2} dx} = e^{-\ln|x - 2|} = e^{\ln|x - 2|^{-1}} = \frac{1}{|x - 2|}$ Since we are looking for a solution that passes through the origin, we can assume $x$ is near 0, so $x - 2 < 0$. Therefore, $|x - 2| = -(x - 2) = 2 - x$, and $\frac{1}{|x-2|} = \frac{1}{2-x}$.

Thus, we take the integrating factor to be $\frac{1}{2-x}$.

Step 3: Multiply the differential equation by the integrating factor and integrate.

Multiplying the differential equation by the integrating factor $\frac{1}{2-x}$, we get: $\frac{1}{2-x}\frac{dy}{dx} - \frac{1}{(x - 2)(2 - x)}y = \frac{x - 2}{2 - x} + \frac{4}{(x - 2)(2 - x)}$ $\frac{1}{2-x}\frac{dy}{dx} + \frac{1}{(2 - x)^2}y = -1 - \frac{4}{(x-2)^2}$ The left-hand side is the derivative of $\frac{y}{2 - x}$ with respect to $x$: $\frac{d}{dx}\left(\frac{y}{2 - x}\right) = -1 - \frac{4}{(x - 2)^2}$ Now, integrate both sides with respect to $x$: $\int \frac{d}{dx}\left(\frac{y}{2 - x}\right) dx = \int \left(-1 - \frac{4}{(x - 2)^2}\right) dx$ $\frac{y}{2 - x} = -x + \frac{4}{x - 2} + C$

Step 4: Apply the initial condition to find the constant of integration.

The curve passes through the origin (0, 0). Substituting $x = 0$ and $y = 0$ into the equation, we get: $\frac{0}{2 - 0} = -0 + \frac{4}{0 - 2} + C$ $0 = -2 + C$ $C = 2$

Step 5: Write the particular solution.

The particular solution is: $\frac{y}{2 - x} = -x + \frac{4}{x - 2} + 2$ $y = (2 - x)\left(-x + \frac{4}{x - 2} + 2\right)$ $y = (2 - x)(-x + 2) + 4$ $y = -2x + x^2 + 4 - 2x + 4$ $y = x^2 - 4x + 8$

Step 6: Check which of the given points lies on the curve.

We need to check which of the given points satisfies the equation $y = x^2 - 4x + 8$.

• (4, 4): $4 = (4)^2 - 4(4) + 8 = 16 - 16 + 8 = 8$. This is false.
• (5, 5): $5 = (5)^2 - 4(5) + 8 = 25 - 20 + 8 = 13$. This is false.
• (5, 4): $4 = (5)^2 - 4(5) + 8 = 25 - 20 + 8 = 13$. This is false.
• (4, 8): $8 = (4)^2 - 4(4) + 8 = 16 - 16 + 8 = 8$.

However, if there was a typo in the options and (4, 8) was an option, this would satisfy the equation. But since only (A), (B), (C), and (D) are options, let's re-examine our derivation.

In Step 1, we have: $\frac{dy}{dx} = (x - 2) + \frac{y + 4}{x - 2}$ $\frac{dy}{dx} - \frac{1}{x - 2}y = x - 2 + \frac{4}{x-2}$ The integrating factor is $\frac{1}{2-x}$. Multiplying both sides by it we have: $\frac{1}{2-x}\frac{dy}{dx} - \frac{1}{(x - 2)(2 - x)}y = \frac{x - 2}{2 - x} + \frac{4}{(x - 2)(2 - x)}$ $\frac{1}{2-x}\frac{dy}{dx} + \frac{1}{(2 - x)^2}y = -1 - \frac{4}{(x-2)^2}$ $\frac{d}{dx}(\frac{y}{2-x}) = -1 - \frac{4}{(x-2)^2}$ Integrating both sides, we get $\frac{y}{2-x} = -x + \frac{4}{x-2} + C$ Since the curve passes through (0, 0), we have $0 = 0 - 2 + C \implies C = 2$ Thus, $\frac{y}{2-x} = -x + \frac{4}{x-2} + 2 = \frac{-x(x-2) + 4 + 2(x-2)}{x-2} = \frac{-x^2 + 2x + 4 + 2x - 4}{x-2} = \frac{-x^2 + 4x}{x-2} = \frac{-x(x-4)}{x-2}$ $y = \frac{-x(x-4)(2-x)}{x-2} = x(x-4)$ Now, we have $y = x^2 - 4x$.

Let's check the options: (A) (4, 4): $4 = 4^2 - 4(4) = 16 - 16 = 0$. False (B) (5, 5): $5 = 5^2 - 4(5) = 25 - 20 = 5$. True (C) (5, 4): $4 = 5^2 - 4(5) = 25 - 20 = 5$. False (D) (4, 5): $5 = 4^2 - 4(4) = 16 - 16 = 0$. False

There must be an error in the problem statement or the options. If the differential equation was $\frac{dy}{dx} = \frac{x^2 - 4x + y}{x - 2}$ and the curve passes through the origin, then $y = x^2 - 4x$. If the options were (A) (4, 0) (B) (5, 5) (C) (5, 4) (D) (4, 5) Then option (B) would be correct.

Given the answer is (4, 4), there must be an error in the initial problem statement. The correct answer given is (A), so let's assume the curve passes through (4, 4). $4 = 16 - 16 + C \implies C = 4$. Thus, $y = x^2 - 4x + 4 = (x-2)^2$. $\frac{dy}{dx} = 2(x-2)$. $2(x-2) = \frac{x^2 - 4x + y + 8}{x-2} \implies 2(x-2)^2 = x^2 - 4x + y + 8 \implies 2(x^2 - 4x + 4) = x^2 - 4x + y + 8$ $2x^2 - 8x + 8 = x^2 - 4x + y + 8 \implies y = x^2 - 4x$. This contradicts that $y = (x-2)^2$.

Let's try the equation $y = x^2 - 4x + 8$. Then $\frac{dy}{dx} = 2x - 4$. $2x - 4 = \frac{x^2 - 4x + x^2 - 4x + 8 + 8}{x - 2} = \frac{2x^2 - 8x + 16}{x-2} = \frac{2(x^2 - 4x + 8)}{x - 2}$. This doesn't seem right.

If the correct answer is (4, 4), then we need $y = x^2 - 4x + C$. When $x = 0, y = 0$, so $C = 0$. Then $y = x^2 - 4x$. This gives (5, 5) as a solution. However, (4, 4) is the given answer.

If we assume $y=x$, then $\frac{dy}{dx} = 1$. $1 = \frac{x^2 - 4x + x + 8}{x - 2} = \frac{x^2 - 3x + 8}{x - 2}$. $x - 2 = x^2 - 3x + 8 \implies x^2 - 4x + 10 = 0$.

The problem statement is incorrect. Given the options and the answer, it is not possible to obtain y = x^2 - 4x + 8.

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when finding the integrating factor and performing the integration.
• Algebraic Manipulation: Double-check your algebraic manipulations to avoid errors. Factoring and simplification are crucial.
• Integrating Factor: Make sure to multiply every term in the differential equation by the integrating factor before integrating.

Summary

The problem asks us to solve a first-order linear differential equation with an initial condition. We identified the equation type, found the integrating factor, and solved for the general solution. Then, we used the initial condition (passing through the origin) to find the particular solution. Finally, we checked which of the given points satisfied the equation. However, after careful derivation, the correct answer does not match the options given. There seems to be an error in the original problem statement or options. Assuming the equation derived, $y=x^2-4x+8$, the option (4, 8) would have been the correct answer. Given that the provided answer is (4, 4), the equation should be $y=x^2-4x$, and option (5, 5) should be the right answer.

Final Answer

The problem statement is flawed. Assuming the equation $y = x^2 - 4x + 8$ then there are no correct options. Assuming the correct equation is $y=x^2-4x$, then the correct option is (B) (5, 5). Since the correct answer is given as (A) (4, 4) without further changes, we can conclude there is an error.