Can you help solve these problems
Problem 1. Find the ordinary differential equation satisfied by the given family of curves. In all the exercises, a, b and c are arbitrary constants. (2) e2 +2c re 20 (3) y = z Problem 2. Verify that the ODEs are h 0 s and solve them. (2) dy-2+y2 Problem 3. Verify that the ODEs are exact and solve them. (I) (t2 +2,2)dt+(2tr-z)dz = 0 (2) (2x sin y + y3e") dr + (2.2 cos y + 3y2e)dy = 0 Problem 4. Solve the given Bernoulli equations. (1) dy = ycot(2) + y3 cc(2) (45)2 2dz+ytan(x)= ys (2) cos(z) (3) y + 2 y =-y22 2 cos(z) Problem 5. Consider the second-order linear ODE xy"-(z + 1)y' + y-x2, x 0, where the prime stands for differentiation with respect to r (a) Verify that yi (z) = x + 1 solves the associate homogeneous equation. (b) Solve the associated homogeneous equation using reduction of order. (c) Find the general solution of the nonhomogeneous ODE using variation of parameters. Problem 6. Find the general solution of the O (i) y" + 4y = cos(2x) (ii) y" + 2y +y e (ii) g"y 0. DEs.
