Question

(1 point) A first order linear equation in the form y' + p(x)y = f(x) can be solved by finding an integrating factor μ(x) = exp ( (1) Given the equation y, +-= 7x4 find μ(x) (2) Then find an explicit general solution with arbitrary constant C p(x) dx (3) Then solve the initial value problem with y(1) = 2

Answer 1

y' + 2/x y = 7x^4 b(x) = 2/x, f(x) = 7x^4 u(x) = u^integral b(x) dx = u^2 integral 1/x dx = u^2 log x = x^2 multiplying by u(x) therefore x^2 y' + 2xy = 7x^6 implies dD/dx(x^2 y) = 7x^6 implies x^2 y = integral 7x^6 dx + c implies x^2 y = x^7 + c implies y = x^5 + c x^-2 because y(1) = 2 implies 2 = 1 + c implies c = 1 therefore y = x^5 + x^-2.