Question

Find the general solution to the differential equation put the problem in standard form. find the integrating factor, . find . use c as the unknown constant.

Answer 1

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

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

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

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

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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.