SUMMARY
The discussion centers on verifying the solution to the ordinary differential equation (ODE) given by the expression (x+2y-4)dx - (2x-4y)dy=0. The user's solution is ln[4(y-1)^2+(x-2)^2] + 2arctan((x-2)/(2y-2))=C, while the textbook's answer is ln[4(y-1)^2+(x-2)^2] - 2arctan((2y-2)/(x-2))=C. The discrepancy is attributed to the use of the identity tan(a) = 1/tan(π/2 - a), confirming that both solutions are equivalent except for a constant. Initial conditions were also discussed but not specified.
PREREQUISITES
- Understanding of ordinary differential equations (ODEs)
- Familiarity with logarithmic and trigonometric identities
- Knowledge of the arctangent function and its properties
- Basic calculus concepts related to integration and constants of integration
NEXT STEPS
- Study the properties of the arctangent function and its identities
- Explore methods for solving ordinary differential equations
- Review integration techniques involving logarithmic functions
- Examine the role of initial conditions in determining unique solutions to ODEs
USEFUL FOR
Students studying differential equations, mathematics educators, and anyone interested in verifying solutions to ODEs and understanding trigonometric identities.
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