Use integration to find the general solution to the differential equation .

Question

Use integration to find the general solution to the differential equation  . ​

Quizplus · Accepted Answer

The differential equation 11eb0d3b_9894_019a_9790_59107b082a3f_TB1243_11 can be written as: dy/dx = 2x + 3y We can separate the variables to obtain: dy/ (2x + 3y) = dx Integrating both sides, we get: (1/3) ln|2x + 3y| = x + C where C is the constant of integration. Exponentiating both sides, we obtain: |2x + 3y| = e^(3x+3C) Since the absolute value of 2x + 3y can take two values depending on the sign of 2x + 3y, we consider two cases: Case 1: 2x + 3y > 0 In this case, we have: 2x + 3y = e^(3x+3C) Solving for y, we get: y = (1/3) (e^(3x+3C) - 2x) Case 2: 2x + 3y < 0 In this case, we have: -2x - 3y = e^(3x+3C) Solving for y, we get: y = (1/3) (-e^(3x+3C) - 2x) Therefore, the general solution is: y = (1/3) (e^(3x+3C) - 2x) if 2x + 3y > 0 y = (1/3) (-e^(3x+3C) - 2x) if 2x + 3y < 0

Use Integration to Find the General Solution to the Differential