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dy/dx + 2xy=x
use bernoulli therorem
hint: this problem is in the form of dy/dx +p(x)y=Q(x)
find integrating factor (IF) then use general formula i.e y*IF= integration of IF *Q(x)

To solve the differential equation

[
\frac{dy}{dx} + 2xy = x
]

using Bernoulli’s theorem, we first recognize that this equation is linear and can be rewritten in the standard form:

[
\frac{dy}{dx} + p(x)y = Q(x)
]

where ( p(x) = 2x ) and ( Q(x) = x ).

Step 1: Find the Integrating Factor (IF)

The integrating factor is given by:

[
IF = e^{\int p(x) , dx} = e^{\int 2x , dx}
]

Calculating the integral:

[
\int 2x , dx = x^2
]

Thus, the integrating factor is:

[
IF = e^{x^2}
]

Step 2: Multiply the entire equation by the Integrating Factor

We multiply the entire differential equation by ( e^{x^2} ):

[
e^{x^2}\frac{dy}{dx} + 2xe^{x^2}y = xe^{x^2}
]

Step 3: Recognize the Left Side as a Derivative

The left-hand side can be recognized as the derivative of a product:

[
\frac{d}{dx}(e^{x^2}y) = xe^{x^2}
]

Step 4: Integrate Both Sides

Integrating both sides with respect to ( x ):

[
\int \frac{d}{dx}(e^{x^2}y) , dx = \int xe^{x^2} , dx
]

The left side simplifies to:

[
e^{x^2}y
]

For the right side, we use substitution. Let ( u = x^2 ), then ( du = 2x , dx ), or ( dx = \frac{du}{2x} ). Hence:

[
\int xe^{x^2} , dx = \frac{1}{2} \int e^u , du = \frac{1}{2}e^{u} + C = \frac{1}{2}e^{x^2} + C
]

Step 5: Write Down the Integrated Equation

Thus we have:

[
e^{x^2}y = \frac{1}{2}e^{x^2} + C
]

Step 6: Solve for ( y )

Now, we can solve for ( y ):

[
y = \frac{1}{2} + Ce^{-x^2}
]

Conclusion

The general solution to the differential equation ( \frac{dy}{dx} + 2xy = x ) is:

[
y = \frac{1}{2} + Ce^{-x^2}
]

where ( C ) is an arbitrary constant determined by initial conditions if they are given.