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Solve the differential equation.

$ 4x^3y + x^4y' = \sin^3x $

$$

y=\frac{\cos ^{3} x-3 \cos x+3 C}{3 x^{4}}

$$

Differential Equations

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Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

in order to solve this equation, the first step would be to divide by X to the power of four in order to get why one by itself. So we have why Prime plus four wide divide by axe is sign cubed of acts divided by axe to the fourth. Okay, now we can do the second stuff, which would be the integrating factor. So eat the integral of four over Axe DX. Okay, we need to integrate this. This gives us eat to the four natural lock of ex. However, this can actually be written as eed to the natural log of axe to the fourth. Now, remember, each the natural log is just one there for the solution to this is simply X to the fourth. Therefore, we now know we have acts to the power of four times. Why is the integral of one minus co sign squirt of acts? Times sine acts d x. Because remember, we now have to integrate the right side because eventually gonna end up with why is something so in this context, you can use interested by parts you can use u substitution if we substitute you as next to the four. That's the power of four times why again, there's multiple different ways of how you can substitute this. There's just this is just one possible example. We have X to the power four times Why is co sign cubed over three times acts minus co sign acts plus C. Now, if we divide each of these terms by extra power for because again, we need to get why by itself we can't have any coefficient on the why besides one. Then we end up with our solution, which is why is co sign Cubed of Acts minus three Coastline ax plus three times are constant and then all divided by three acts to the fourth, and this is an example of how you can leave the answer.