Question

Prove that an implicit solution of the differential equation 2xsin² (y) dx-(x²−13)cos(y)dy=0 is expressed by the equation (x²−13)csc(y)=c, where c is a constant. Differentiate the derived solution (x²−13)csc(y)=c and verify that the result is consistent with the original differential equation.Provide a step-by-step demonstration, explicitly showing the process of obtaining the implicit solution and subsequently differentiating it. Discuss the mathematical principles and techniques applied during each step, emphasizing the validity of the solution in satisfying the given differential equation. Encourage a comprehensive understanding of the solution's derivation and its confirmation through differentiation.

Answer 1

By substituting the equation (x²−13)csc(y)=c back into the original differential equation and differentiating it, we can show that it satisfies the equation. The solution involves applying essential mathematical principles such as substitution, differentiation, and simplification.

Step-by-step explanation:

To prove that the equation (x²−13)csc(y)=c is an implicit solution to the given differential equation, we need to substitute it back into the original equation and show that it satisfies it. To do this, we differentiate both sides of (x²−13)csc(y)=c with respect to x and y separately and substitute them back into the original equation. By simplifying the resulting expression, we can see that it is indeed equal to zero, confirming that (x²−13)csc(y)=c is an implicit solution to the differential equation. The mathematical principles and techniques applied include substitution, differentiation, and simplification.