Solve Differential equation

Question

Solve ( x+y+1)dy/dx = 1

Answer 1

suppose z = x+y+1 say

1+dy/dx = dz/dx, so... dy/dx = (dz/dx)-1

these value put in the problem... so....

z(dz/dx - 1) = 1  so.... dz/dx -1 = 1/z

dz/dx = 1+1/z = z+1/z

(z/z+1)dz = dx

( 1- 1/Z+1 )dz = dx

both sides integrel : z-log(z+1) = x+c

answer is: y-log (x+y+2) = c

Answer 2

. Consider the differential equation
(t sin(t) + cos(t))y − t cos(t)y + cos(t)y = 0. (2)
(a) Find and solve a differential equation satisfied by the Wronskian W(y1,y2)(t) of a fundamental set of solutions y1(t),y2(t) for (2).
(b) Check that y1(t) = t is a solution and find another linearly independent solution.
(c) Write down the general solution, and find the solution with (y(0), y′(0)) = (1, 1).