((3y^2)dx/(x^2+3x)) +( ( 2y ln ((5x)/(x+3))+ 3siny)dy=0
Please help me in solving this problem using exact differential equation.
Thanks in advance.
you can start by noting that if
u = ln(5x/(x+3))
du/dx = 3/(x^2+3x)
The rest should be not so hard now.
To solve this problem using exact differential equation, we can follow these steps:
Step 1: Determine if the equation is exact.
A differential equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if and only if ∂M/∂y = ∂N/∂x.
Let's differentiate M(x, y) with respect to y:
∂M/∂y = 0
Now, let's differentiate N(x, y) with respect to x:
∂N/∂x = 2y(5/(5x/x+3) - 1/(x+3)^2) + 3cosy
Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact.
Step 2: Multiply the entire equation by an integrating factor.
To make the equation exact, we need to find an integrating factor, μ(x, y), which satisfies the following condition: μ(x, y)[∂M/∂y - ∂N/∂x] = μ(x, y)(0 - 2y(5/(5x/x+3) - 1/(x+3)^2) - 3cosy).
Simplifying the above expression, we have:
μ(x, y)(- 2y(5/(5x/x+3) - 1/(x+3)^2) - 3cosy) = 0
Step 3: Solve for the integrating factor.
We equate the above equation to zero and solve for μ(x, y):
- 2y(5/(5x/x+3) - 1/(x+3)^2) - 3cosy = 0
Simplify the equation further:
- 10y/(5x/x+3) + 2y/(x+3)^2 = 3cosy
Now, solve this equation for μ(x, y):
μ(x, y) = (- 10y/(5x/x+3) + 2y/(x+3)^2) / 3cosy
Step 4: Multiply the original equation by the integrating factor μ(x, y).
Multiply both sides of the original equation by μ(x, y) to obtain the exact differential equation:
μ(x, y)[(3y^2)dx/(x^2+3x)] + μ(x, y)[(2y ln((5x)/(x+3)) + 3siny)dy] = 0
Step 5: Check if the equation is now exact.
Repeat Step 1 to check if the equation is now exact. If it is, move on to the next step. If not, recheck your calculations.
Step 6: Integrate both sides of the equation.
Integrate each term separately with respect to its corresponding variable. Treat the other variable as a constant.
Integrating the left-hand side, we have:
∫[μ(x, y)((3y^2)dx/(x^2+3x))] + ∫[μ(x, y)((2y ln((5x)/(x+3)) + 3siny)dy)] = ∫[0] [Integrating zero will give us a constant C.]
Step 7: Simplify the equation and solve for the constant C.
Simplify the left-hand side and solve for the constant C using the given boundary conditions or additional information.
Step 8: Write the final solution.
Once you have the value of the constant C, you can write the final solution to the differential equation with the given boundary conditions or additional information.