Dy/dx=x+x^3/y
plz show your work
do you mean (x+x^3)/y or x + x^3/y ?
It makes a difference
Yes sir...sorry for the typo
To solve the differential equation dy/dx = x + x^3/y, we can use separation of variables. The basic idea behind separation of variables is to move all terms involving y to one side of the equation and move the terms involving x to the other side.
1. Start by rearranging the equation to have all terms involving y on one side:
dy/dx = x + x^3/y
Multiply both sides by y:
y * dy/dx = xy + x^3
2. Next, move all terms involving x to the other side:
y * dy = (xy + x^3) * dx
3. Now, we can separate the variables by dividing both sides of the equation by (xy + x^3):
(1/y) * dy = dx
4. Integrate both sides with respect to their respective variables:
∫(1/y) * dy = ∫dx
The integral of (1/y) with respect to y is ln|y| (natural logarithm of absolute value of y) and the integral of dx is x.
Therefore, we have:
ln|y| = x + C, where C is the constant of integration.
5. To solve for y, we can take the exponential of both sides:
e^(ln|y|) = e^(x + C)
y = e^(x + C)
Since e^C is also a constant, we can rewrite it as K, where K = e^C:
y = Ke^x
And that completes the solution to the differential equation.