Question

Use the method for solving equations with linear coefficients to solve the following differential equation. (2x-y) dx +(4x y 3) dy 0 Ignoring lost solutions, if any, an implicit solution in the form F(xy :cis |-C where C is an arbitrary constant. Type an expression using x and y as the variables.)Use the method for solving equations with linear coefficients to solve the following differential equation. (2x-y) dx +(4x y 3) dy 0 Ignoring lost solutions, if any, an implicit solution in the form F(x y)=C is ______C where C is an arbitrary constant. Type an expression using x and y as the variables.)

Answer 1

To solve the differential equation with linear coefficients, attempt to separate variables, integrate each side with respect to its variable, combine constants into a single C, and represent the solution in the form F(x, y) = C.

Step-by-step explanation:

To solve the given differential equation (2x-y) dx + (4x - y + 3) dy = 0 using the method of solving equations with linear coefficients, we must identify a solution that incorporates an arbitrary constant C. This type of equation can often be directly integrable if it is exact or can be made exact via an integrating factor. However, since no initial solution method is directly provided in the problem statement, we will instead look for a solution that satisfies the differential equation form F(x, y) = C.

• First, we attempt to separate variables and integrate, which is the standard approach to solving first-order linear differential equations.
• If separable, integrate each side with respect to its corresponding variable.
• Combine the integration constants into a single constant C.
• Write the final solution in the implicit form F(x, y) = C.