In Exercises 1–22, solve the differential equation.
x dy...

Question

In Exercises 1–22, solve the differential equation.

x dy + (3y - x⁻² cos x) dx = 0, x > 0

Accepted Answer

Hello. In this video, we are going to be finding the general solution to the given differential equation. The differential equation given to us is x multiplied by DY plus the quantity 4Y minus x 3 multiplied by cosine of X, and this entire quantity is multiplied by DX and this is going to equal to 0. So in order to start solving for the general solution, we need to first identify our differential DYD X. In order to do this, we will divide every term in this equation by dx. That will allow us to rewrite the differential equation as x multiplied by DYDX plus 4 Y minus x to the 3 cosine of x equal to 0. Now what we want to do is we want to go ahead and simplify this into a first-order linear differential equation. We can start by adding x to the 3rd cosine of x to the right-hand side of the equation. That will leave us with x multiplied by D Y D X plus 4 Y is equal to positive x to the negative 3 power multiplied by cosine of x. And we want to go ahead and give the differential DYDX a coefficient of 1. So in order to do this, we will multiply this entire equation by X to the negative 1st. That will simplify the differential equation to be D Y D X plus 4 divided by X multiplied by Y, and this is going to equal to x to the 4th multiplied by cosine of X. So what we have here is, again, a first-order linear differential equation. Now, the general form of a first order linear differential equation is DYDX plus a function of X multiplied by Y equal to a function of X which we would call Q of X. Now, the right hand side of this equation is going to be Q of X, and 4 divided by X is a function in front of Y, so that is going to be P of X in the first order linear differential equation. Now in order to simplify this equation, we need to identify our integrating factor. Our integrating factor is defined as m equal to e raised the power of the integral of p of x. Now again, we identified p of x as 4 divided by x. So if we plug this into the identity of the integrating factor, we get e raised the power of the integral of 4 divided by x. This will further simplify to be e raised to the power of 4 multiplied by the natural logarithm of x. Now we can further simplify this by using our natural logarithm properties. We can rewrite this integrating factor to be e raced to the power of the natural logarithm of x 4, and this will simplify to be x 4. Now, what we are going to do with this integrating factor is we're going to take this and multiply it to the entirety of our differential equation. So, that is going to simplify our differential equation to be x 4 multiplied by DYDX plus 4 X cubed equal 4 X cubed Y. Equal to cosine of x. No. If we take a look at the left hand side of this equation, the left hand side of the equation is equivalent to the derivative of the product of two terms, and that product is going to be x 4 multiplied by y. So we can go ahead and rewrite the left hand side equation as this derivative, that is going to be the derivative with respect to x of x 4 multiplied by y. And that is going to equal the cosine of x. Now, what we need to do is we need to remove the differential symbol. In order to do this, we will integrate both sides of this equation with respect to x. That is going to leave us with x 4 multiplied by y equal to sin of x plus the generic constant c. Now the last thing we need to do is isolate for y on the left hand side of the equation, and to do that, we will divide every term in this equation by x 4. That will leave us with y is equal to 1 divided by x to the 4th multiplied by sin of x. Plus the generic constant c. This is going to be the simplest form of our general solution. And with that being said, the overall solution to this problem is going to be option A. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

In Exercises 1–22, solve the differential equation.x dy + (3y - x⁻² cos x) dx = 0, x > 0

Watch next

In Exercises 1–22, solve the differential equation.
x dy + (3y - x⁻² cos x) dx = 0, x > 0