dx%2B(2x%5E2%2B3xy)dy%3D0.jpeg "(3x^2+6xy+3y^2)dx+(2x^2+3xy)dy=0")
Solving the Differential Equation (3x^2+6xy+3y^2)dx+(2x^2+3xy)dy=0
The given differential equation is:
(3x^2+6xy+3y^2)dx+(2x^2+3xy)dy=0
This equation is a first-order homogeneous differential equation, which can be solved by using the following steps:
1. Check for Homogeneity
A differential equation is homogeneous if it can be written in the form:
M(x,y)dx + N(x,y)dy = 0
Where M(x,y) and N(x,y) are homogeneous functions of the same degree. In our case:
- M(x,y) = 3x^2 + 6xy + 3y^2
- N(x,y) = 2x^2 + 3xy
Both M(x,y) and N(x,y) are homogeneous functions of degree 2. Therefore, the given equation is homogeneous.
2. Substitute y = vx
To solve the homogeneous equation, we substitute y = vx where v is a function of x. This substitution will transform the given equation into a separable differential equation.
Substituting y = vx and dy = vdx + xdv into the given equation, we get:
(3x^2 + 6x(vx) + 3(vx)^2)dx + (2x^2 + 3x(vx))(vdx + xdv) = 0
Simplifying the equation:
(3x^2 + 6x^2v + 3x^2v^2)dx + (2x^2v + 3x^2v^2)dx + (2x^3v + 3x^3v^2)dv = 0
Combining like terms:
(5x^2 + 9x^2v + 3x^2v^2)dx + (2x^3v + 3x^3v^2)dv = 0
3. Separate the Variables
Now, we can separate the variables by dividing both sides by x^2(5 + 9v + 3v^2) and (2xv + 3xv^2):
(dx/x) + (2v + 3v^2)/(5 + 9v + 3v^2)(dv) = 0
4. Integrate both sides
Integrating both sides of the equation:
ln|x| + (1/3)ln|5 + 9v + 3v^2| = C
Where C is the constant of integration.
5. Substitute back y = vx
Finally, we substitute back y = vx to get the solution in terms of x and y:
ln|x| + (1/3)ln|5 + 9(y/x) + 3(y/x)^2| = C
Solution
The general solution to the given differential equation is:
ln|x| + (1/3)ln|5x^2 + 9xy + 3y^2| = C
This solution can be written in various forms depending on the specific needs of the problem.
This solution represents a family of curves, each with a different constant C.