dy/dx=1+2x-y/x^2
find the solution of differential equation given that when x=1,y=2

To solve the differential equation \(\frac{dy}{dx} = 1 + \frac{2x - y}{x^2}\) with the initial condition \(y(1) = 2\), let's go through the steps.

### Step 1: Rewrite the Equation
We start by isolating terms involving \(y\):
\[
\frac{dy}{dx} = 1 + \frac{2x}{x^2} - \frac{y}{x^2}
\]
Simplify \(\frac{2x}{x^2}\) to get \(\frac{2}{x}\):
\[
\frac{dy}{dx} = 1 + \frac{2}{x} - \frac{y}{x^2}
\]

### Step 2: Rearrange to a Linear Form
Rewrite the equation as:
\[
\frac{dy}{dx} + \frac{y}{x^2} = 1 + \frac{2}{x}
\]
This is now a first-order linear differential equation in the form:
\[
\frac{dy}{dx} + P(x) y = Q(x)
\]
where \(P(x) = \frac{1}{x^2}\) and \(Q(x) = 1 + \frac{2}{x}\).

### Step 3: Find the Integrating Factor
The integrating factor \(I(x)\) is given by:
\[
I(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x^2} \, dx}
\]
Calculating the integral:
\[
\int \frac{1}{x^2} \, dx = -\frac{1}{x}
\]
Thus, the integrating factor is:
\[
I(x) = e^{-\frac{1}{x}} = \frac{1}{e^{1/x}}
\]

### Step 4: Multiply Through by the Integrating Factor
Multiplying both sides of the differential equation by \(\frac{1}{e^{1/x}}\):
\[
\frac{1}{e^{1/x}} \frac{dy}{dx} + \frac{y}{x^2 e^{1/x}} = \left(1 + \frac{2}{x}\right) \frac{1}{e^{1/x}}
\]

### Step 5: Solve

dy/dx=1+2x-y/x^2. Find the solution of the differential equation given that when x=1, y=2.

Explore the step-by-step solution of the differential equation dy/dx = 1 + 2x - y/x^2 with the initial condition y(1) = 2. Learn key concepts in linear differential equations, integrating factors, and solving techniques for first-order differential equations. Ideal for undergraduate students.

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