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34kurt
9 months ago
11

Find the general solution to the differential equation put the problem in standard form. find the integrating factor, . find . u

se c as the unknown constant.
Mathematics
1 answer:
antiseptic1488 [7]9 months ago
3 0

y = \frac{-x}{2 } + \frac{1}{4}  + ce^-{2x} is  the general solution to the differential equation put the problem in standard form.

What is meant by integrating factor?

  • A function called an integrating factor is used to solve differential equations in mathematics. It is a function that can be made integrable by multiplying it by an ordinary differential equation.
  • Ordinary differential equations are typically solved using this method. This factor is also applicable to multivariable calculus.

x² + 2xy + x . dy/dx = 0

x dy/dx  + 2xy = - x^{v}

\frac{dy}{dx} =+ 2y = -x

left u(x) = e^{\int\limits^_ {} } 2dx     = e^{2x}

integration  factor p(x) = u(x) = e^{2x}

Now ,multiply  e^{2x} both sides

e^{2x}  ( \frac{dy}{dx} + 2y )  = -x . e^{2x}

e^{2x} \frac{dy}{dx} + 2e^{2x} . y = -x e^{2x}

\frac{d}{dx} (e^{2x} .y) = -x e^{2x}

integrate both sides

∫ d .( e^{2x} .y )     = ∫-xe^{2x} .dx

e^{2x} .y = -\frac{e^{2x}}{2} . x + \frac{e^{2x}}{4} + c

y = \frac{-x}{2 } + \frac{1}{4}  + ce^-{2x}

Learn more about integrating factor

brainly.com/question/25527442

#SPJ4

The complete question is -

Find the general solution to the differential equation 2 dy X+ + 2xy + x dx = 0 Put the problem in standard form. Find the integrating factor, p(x) = 2x - Find y(x) Use C as the unknown constant.

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