In Exercises 1–22, solve the differential equation.

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Question

In Exercises 1–22, solve the differential equation.

x dy - (x⁴ - y) dx = 0

Accepted Answer

Welcome back everyone. Find the general solution of xDY minus x cubed plus y dx equals 0 for x greater than 0. For this problem, let's begin by distributing, so we're going to get x multiplied by Dy minus. X cubed. D x minus y dx equals 0. Now let's go ahead and divide both sides by dx so that we get x multiplied by dyy divided by dx minus x cubed minus y is equal to 0. Now what we're going to do is simply write it in a standard form of the first order linear differential equation. Specifically, let's go ahead and add x cubed to both sides so that we get X D Y divided by DX. Minus y equals x cubed and now let's divide both sides by x. So we get DY divided by DX minus. 1 divided by x multiplied by y equals x cubed divided by x gives us x 2. And now we are ready to find the integrating factor. Let's notice that the function in front of y is -1 divided by x. So the integrating factor of mu is going to be e raised to the power of the integral of -1 divided by x d x. Now let's go ahead and integrate that's e to the power of negative of l and of x. And now using the properties of logarithms, we get e to the power of ln of 1. Divided by X, which is 1 divided by X, that's our integrating factor. Let's now multiply both sides of our equation by the integrating factor to get 1 divided by x, d y divided by d x. 1 divided by x 2 multiplied by y equals x. And now let's notice that on the left side we have the derivative d divided by dx of the product between y and 1 divided by x or simply y divided by x. On the right hand side we have x. And now if we multiply both sides by dx, we get d of y divided by x equals x dx, which allows us to integrate both sides. On the left side we have the integral of the differential which simply gives us y divided by x. On the right side we can apply the power rule to get x 2 divided by 2 plus a constant of integration c. And from here let's multiply both sides by x so that our final solution is y equals. X cubed. Divided by 2 + C X. That's our final answer, and thank you for watching.

In Exercises 1–22, solve the differential equation.x dy - (x⁴ - y) dx = 0

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In Exercises 1–22, solve the differential equation.

x dy - (x⁴ - y) dx = 0