how do I solve 3xy^3dy/dx=y^2x^2+5x^3y+y^4
To solve the given differential equation, 3xy^3(dy/dx) = y^2x^2 + 5x^3y + y^4, we need to separate the variables and integrate both sides.
1. Start by rearranging the equation to have all terms involving "y" on the left side and all terms involving "x" on the right side:
3xy^3(dy/dx) - y^2x^2 - 5x^3y - y^4 = 0
2. Factor out y^3 from the left side:
y^3(3x(dy/dx) - yx^2 - 5x^3 - y^3) = 0
3. We now have a product of two factors that are individually equal to zero. Set each factor equal to zero:
y^3 = 0 (Factor 1)
3x(dy/dx) - yx^2 - 5x^3 - y^3 = 0 (Factor 2)
4. Solve Factor 1:
y^3 = 0
This means that y = 0 is a possible solution.
5. Solve Factor 2 for the differential equation:
3x(dy/dx) - yx^2 - 5x^3 - y^3 = 0
To solve this first-order linear differential equation, we rearrange it as follows:
3x(dy/dx) = yx^2 + 5x^3 + y^3
Divide both sides by (yx^2 + 5x^3 + y^3):
(dy/dx) = (yx^2 + 5x^3 + y^3) / (3x)
6. Now, we can integrate both sides with respect to x to find the general solution:
∫(dy/dx) dx = ∫[(yx^2 + 5x^3 + y^3) / (3x)] dx
Integrating the left side gives:
∫(dy/dx) dx = y + C1 (where C1 is the constant of integration)
For the right side, we need to perform a partial fraction decomposition:
∫[(yx^2 + 5x^3 + y^3) / (3x)] dx = ∫(y/3) dx + 5∫(x^2/3) dx + ∫(y^3/3x) dx
Integrating each term yields:
(1/3)∫y dx + (5/3)∫x^2 dx + (1/3)∫y^3/x dx
Simplifying further:
(1/3)(y∫1 dx) + (5/3)(x^2∫1 dx) + (1/3)∫y^3/x dx
(1/3)(yx) + (5/3)(x^3/3) + (1/3)∫y^3/x dx
Combining all terms, the right side can be written as:
(yx + 5x^3/3 + (1/3)∫(y^3/x) dx)
Thus, the integrated equation becomes:
y + C1 = yx + 5x^3/3 + (1/3)∫(y^3/x) dx
7. This is the general solution to the differential equation. However, if initial conditions are given, you can substitute them into the equation and solve for the constant C1.