Dy/dx=(X+x^3)/y
help show working
y dy = x dx + x^3 dx
y^2/2 = x^2/2 + x^4/4 + c
It might be easier to break-up the fraction then take the derivative of each:
(X+x^3)/y ==> x/y + x^3/y
Dy/dx = 1/y + 3x^2/y = 3x^2/y
What Damon did was the anti-derivative or indefinite integral. Dy/dx is Leibniz's notation which means you are taking the derivative of y with respect to x. It is the prime notation for the derivative of a function. The second answer would be correct. 3x^2/y is the derivative of your function.
I think Damon is right on this, otherwise, the equals sign is meaningless.
To find the solution to the differential equation dy/dx = (x + x^3)/y, we can use a method called separation of variables. Here's how you can show the working:
Step 1: Rearrange the equation
Start by rearranging the equation to separate the variables. Multiply both sides of the equation by y to get rid of the denominator:
y dy = (x + x^3) dx
Step 2: Integrate both sides
Now, integrate both sides of the equation separately with respect to their respective variables. The integral of y dy on the left side will give you (1/2)y^2, and the integral of (x + x^3) dx on the right side will give you (1/2)x^2 + (1/4)x^4 + C, where C is the constant of integration.
(1/2)y^2 = (1/2)x^2 + (1/4)x^4 + C
Step 3: Solve for y
To get the solution in terms of y, isolate y on one side of the equation. Start by multiplying both sides by 2 to eliminate the denominator:
y^2 = x^2 + (1/2)x^4 + 2C
Now, take the square root of both sides to get:
y = ±√(x^2 + (1/2)x^4 + 2C)
This is the general solution to the given differential equation.
Step 4: Apply initial conditions (if given)
If you have additional information, such as an initial condition, you can substitute those values into the general solution to find the particular solution that satisfies the given conditions.