# Find the solution of the differential equation #xy'=y+x^2\sinx# that satisfies the given initial conditions below?

##
#y(\pi)=0#

I know it's in #dy/dx+P(x)y=Q(x)# form, but what exactly is the integrating factor?

Like what is "exp" in #I=exp(\int P(x)dx)# ?

(People use this a lot when mentioning the integrating factor, but I learned it as #I(x)=e^(\intP(x)dx)# ...)

I know it's in

Like what is "exp" in

(People use this a lot when mentioning the integrating factor, but I learned it as

Refer to the Explanation.

I think I've allayed your curiosity!

By the way, I'm assuming you are familiar with solving the provided diff.

eqn., so I'm unable to solve it.

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To solve the differential equation (xy' = y + x^2 \sin(x)) with the initial condition (y(1) = 2), you can use the method of integrating factors. First, rewrite the equation in standard form (y' + P(x)y = Q(x)), where (P(x)) is the coefficient of (y) and (Q(x)) is the remaining terms:

[y' - \frac{1}{x}y = x \sin(x)]

Then, identify (P(x) = -\frac{1}{x}) and (Q(x) = x \sin(x)). Next, find the integrating factor (I(x)), which is given by:

[I(x) = e^{\int P(x) , dx}]

Calculate the integral:

[\int -\frac{1}{x} , dx = -\ln|x|]

So, the integrating factor is (I(x) = e^{-\ln|x|} = \frac{1}{|x|}).

Now, multiply both sides of the differential equation by the integrating factor:

[\frac{1}{|x|} y' - \frac{1}{|x|^2}y = \sin(x)]

This can be rewritten as:

[\frac{d}{dx} \left(\frac{y}{|x|}\right) = \sin(x)]

Integrate both sides with respect to (x):

[\frac{y}{|x|} = -\cos(x) + C]

Where (C) is the constant of integration.

Now, multiply both sides by (|x|) to solve for (y):

[y = -|x| \cos(x) + C|x|]

Now, apply the initial condition (y(1) = 2) to find the value of (C):

[2 = -1 \cdot \cos(1) + C \cdot 1]

[C = 2 + \cos(1)]

So, the solution to the differential equation with the given initial condition is:

[y = -|x| \cos(x) + (2 + \cos(1)) |x|]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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