Question

Use the reduction of order method to find the general solution of each of the following equations. One solution of the homogeneous equation is shown alongside each equation.

Answer 1

(A) The given differential equation is y'- 3 = squareroot 10y - 4x + 1 .....(1) putting 10y - 4x + 1 = z^2 then 10 dy/dx - 4 = 2z dz/dx dy/dx = 1/10 (2z dz/dx + 4) from (1) we have 1/10 (2z dz/dx + 4) - 3 = z z/5 dz/dx + 2/5 - 3 = z z/5 dz/dx - 13/5 = z \r\n dz/dx - 13/z = 5 dz/dx = 5 + 13/z = 5z + 13/z z dz/5z + 13 = dx 1/5 (5z + 13 - 13)/5z + 13 dz = dx 1/5 (1- 13/5z - 13)dz = dx Integrating, 1/5 [z - 13/5 log (5z + 13)] = x + c [e is arbitrary constant] 1/5{squareroot 10y - 4x + 1 - 13/5 log (5 squareroot 10y - 4x + 1 + 13)} = x + c which is the required general solution