The solution of the differential equation `(dy)/(dx)=(x+y)/x` satisfying the condition `y""(1)""=""1` is (1) `y""="ln"x""+""x` (2) `y""=""x"ln"x""+""x^2` (3) `y""=""x e(x-1)` (4) `y""=""x"ln"x""+""x`

Options

A. `y=xe^(x-1)`
B. `y=x ln x+x`
C. `y=ln x+x`
D. `y=x ln x+x^(2)`

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{x+y}{x}\) with the condition \(y(1) = 1\), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ \frac{dy}{dx} = \frac{x+y}{x} \]
This can be rewritten as:
\[ \frac{dy}{dx} = 1 + \frac{y}{x} \]

### Step 2: Rearrange to Standard Form
Rearranging gives us:
\[ \frac{dy}{dx} - \frac{y}{x} = 1 \]
This is now in the standard form of a linear differential equation, which is:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \(P(x) = -\frac{1}{x}\) and \(Q(x) = 1\).

### Step 3: Find the Integrating Factor
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int P(x) \, dx} = e^{\int -\frac{1}{x} \, dx} = e^{-\ln|x|} = \frac{1}{x} \]

### Step 4: Multiply the Equation by the Integrating Factor
We multiply the entire differential equation by the integrating factor:
\[ \frac{1}{x} \left( \frac{dy}{dx} - \frac{y}{x} \right) = \frac{1}{x} \]
This simplifies to:
\[ \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = \frac{1}{x} \]

### Step 5: Rewrite the Left Side
The left side can be rewritten as:
\[ \frac{d}{dx}\left(\frac{y}{x}\right) = \frac{1}{x} \]

### Step 6: Integrate Both Sides
Integrating both sides gives:
\[ \int \frac{d}{dx}\left(\frac{y}{x}\right) \, dx = \int \frac{1}{x} \, dx \]
This results in:
\[ \frac{y}{x} = \ln|x| + C \]
where \(C\) is the constant of integration.

### Step 7: Solve for \(y\)
Multiplying through by \(x\) gives:
\[ y = x \ln|x| + Cx \]

### Step 8: Apply the Initial Condition
We apply the initial condition \(y(1) = 1\):
\[ 1 = 1 \cdot \ln(1) + C \cdot 1 \]
Since \(\ln(1) = 0\), we have:
\[ 1 = C \]
Thus, \(C = 1\).

### Step 9: Final Solution
Substituting \(C\) back into the equation gives:
\[ y = x \ln|x| + x \]

### Conclusion
The final solution is:
\[ y = x \ln x + x \]

### Matching with Options
Looking at the provided options, we find that the correct answer is:
(2) \(y = x \ln x + x^2\) is incorrect, the correct answer is \(y = x \ln x + x\).