Question

Please complete both if possible if not please complete # 10. Thank you. Print clearly.

9. Find the solution of the differential equation (3 + 2sin(a2))y' 4x cos(r2) y 2x, such that y(0)-| 10. Find the solution of the differential equation such that y(0) = 1.

Answer 1

(3 + 2 sin (x^2))y^' + 4x cos(x^2)y = 2x (3 + 2 sin (x^2))dy + (4x middot cos (x^2)y -2x)dx = 0 equation (1) M = 4x cos(x^2)y -2x, N = 3 + 2 sin(x^2) partial differential M/partial differential y = partial differential N/partial differential x implies (1) is exact Therefore (1) implies d(3y) + d(2 sin (x^2)y) + d(-x^2) = d(c) on integrating both sides 3y + 2 sin (x^2)y - x^2 = c now, y(0) = 0 implies 0 + 0 -0 = c implies c = 0 3y + 2 sin(x^2)y - x^2 = 0 (1 + e^x)y^' + e^x y = e^x (1 + e^x)dy = e^x(1 - y)dx integral dy/1 - y = integral e^x/1 + e^x dx let 1 + e^x = t implies e^x dx = dt integral dy/1 - y = dt/t ln (1 - y) = ln t + ln c 1 - y = ct = c(1 + e^x) y(0) = 1 implies 1 -1 = c(1 + 1) implies \r\n