Question

The differential equation dxdy=49+49x+21y+21xy has an implicit general solution of the form F(x,y)=K, where K is an arbitrary constnat. In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form F(x,y)=G(x)+H(y)=K Find such a solution and then give the related functions requested. F(x,y)=G(x)+H(y)=

Answer 1

The implicit solution curve for the given differential equation is represented by the function F(x, y) = 7x² + 21xy + 49y + C, where C is an arbitrary constant.

Step-by-step explanation:

To find the implicit solution, we integrate the given differential equation. The separable differential equation dxdy = 49 + 49x + 21y + 21xy can be rearranged as (1/(49 + 21y))dy = dx, making it suitable for separation of variables.

Integrating both sides gives us the implicit general solution:

∫ (1/(49 + 21y)) dy = ∫ dx

The integration yields:

7 ln|49 + 21y| = x + K₁

Solving for y and exponentiating both sides:

|49 + 21y| = e^((x + K₁)/7)

Considering the arbitrary constant K₂ = e^(K₁/7), we get:

49 + 21y = K₂e^(x/7)

Rearranging terms and setting C = 49/K₂, the implicit solution curve becomes:

F(x, y) = 7x² + 21xy + 49y + C

This form represents the solution curve, and the related functions are G(x) = 7x² + C and H(y) = 49y + C, where C is an arbitrary constant.