Solving the Differential Equation ( frac{y sin x - cos x}{y sin x cos x} , dy dx )
This article will guide you step-by-step through solving the differential equation:
1. Simplification and Rearrangement
The given differential equation is:
Step 1: Start with the given equation:
( frac{y sin x - cos x}{y sin x cos x} , dy dx )
Step 2: Simplify the left-hand side of the equation:
( frac{y sin x}{y sin x cos x} - frac{cos x}{y sin x cos x} , dy dx )
( frac{sin x}{cos x} - frac{cos x}{y sin x cos x} , dy dx )
( tan x - frac{cos x}{y sin x cos x} , dy dx )
Step 3: Now, rearrange the terms to make it more understandable:
( tan x , dy - frac{cos x}{y sin x cos x} , dy dx )
( tan x , dy - frac{1}{y sin x} , dy dx )
Step 4: Factor out the common term ( frac{1}{y sin x} ) from the left-hand side:
( tan x , dy - frac{1}{y sin x} , dy dx )
Step 5: Notice that we can combine the terms on the left-hand side:
( frac{y sin x - cos x}{y sin x cos x} , dy dx )
2. Integration and Solving the Exact Equation
The equation is exact because:
Step 6: Verify the exactness of the equation by checking the partial derivatives:
( left( frac{partial}{partial y} right) left( frac{y sin x - cos x}{y sin x cos x} right) cos x )
( left( frac{partial}{partial x} right) left( y sin x cos x right) cos x )
The partial derivatives are equal, confirming the equation is exact.
3. Integration of the Exact Equation
Since the equation is exact, we can integrate both sides:
Step 7: Integrate ( y sin x , dy cos x , dx )
Step 8: Integrate the left-hand side:
( int y sin x , dy frac{y^2 sin x}{2} g(x) )
Step 9: Integrate the right-hand side:
( int cos x , dx sin x h(y) )
Step 10: Combine the results:
( frac{y^2 sin x}{2} g(x) sin x h(y) )
Step 11: Simplify and find a constant ( C ) to combine the constants of integration:
( frac{y^2 sin x}{2} - sin x C )
Step 12: Factor out ( sin x ) from the left-hand side:
( sin x left( frac{y^2}{2} - 1 right) C )
The general solution of the differential equation is:
( boxed{frac{y^2}{2} - 1 frac{C}{sin x}})
Conclusion
The final solution to the differential equation ( frac{y sin x - cos x}{y sin x cos x} , dy dx ) is:
( frac{y^2}{2} - 1 frac{C}{sin x})