How to solve this problem? x dy/dx +2y = sinx/x, y(2)=1classify the equation: linear, nonlinear, separable,exact, homogeneous, or one that requires an integration factor.

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beckden | High School Teacher | (Level 1) Educator

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Our equation is a linear ordinary differential equation of order 1.

These equations are of the form

(dy)/(dx) + f(x)y(x) = g(x).

We can solve them by multiplying by `e^(int f(x)dx)`

It might be easier to show the method if I solve this problem.

In our case `f(x)=2/x` , `g(x) = sin(x)/x^2` .  We got this by dividing both sides by x.

`int f(x) dx = int 2/x dx = 2 ln(x)` ,

We are going to multiply both sides by `e^(2ln(x))=e^(ln(x^2))=x^2`

To get

`x^2(dy)/(dx) + 2xy = sinx`

Now we notice that `(d(x^2y))/(dx) = x^2(dy)/(dx) + 2xy` by the product property.

So our equation is actually

`(d(x^2y))/(dx) = sinx`

Integrating both sides we get

`x^2y = -cosx + C`

Solving for y we get

`y = -(cosx)/(x^2) + C/(x^2)`

Using our intital condition that y(2) = 1

`1 = -(cos(2))/(2^2) + C/(2^2)`

`1+(cos(2))/4 = C/4`

`C = 4 + cos(2)`

So our solution is

`y = -(cos(x))/(x^2) + (4 + cos(2))/x^2`

Hope that helps...

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