Final answer:
To solve the differential equation dy/dx = (y - 14x) + 5, a substitution such as v = y - 14x is used, leading to an integrated form v = 5x + C. Plugging the initial condition y(0) = 0 into the integrated form determines the constant C and gives the solution y = 19x. For finding the integrating factor of (dy/dx) - 3y = 0, the integrating factor is exp(-3x).
Step-by-step explanation:
Part A: Solving the Differential Equation
To solve the differential equation dy/dx = (y - 14x) + 5 with the initial condition y(0) = 0, we first set up the equation to standard form by removing the '14x' to the right side:
dy/dx - y = -14x + 5
This is a first-order linear differential equation. We can solve it using an integrating factor, but since the question asks for a substitution, we consider a change of variables, such as v = y - 14x. By differentiating v with respect to x, we get dv/dx = dy/dx - 14. Substituting the original equation into this gives dv/dx = 5, which is easily integrated to v = 5x + C. To find the constant C, we use the initial condition y(0) = 0, which implies v = -14(0) = 0, so C = 0. Therefore, v = 5x, and substituting back for y gives y = 5x + 14x = 19x.
Part B: Finding the Integrating Factor
To find the integrating factor of the differential equation (dy/dx) - 3y = 0, we can recognize it as a standard linear first-order differential equation of the form dy/dx + p(x)y = q(x), where p(x) = -3 and q(x) = 0. The integrating factor, μ, is given by the exponential of the integral of p(x). Hence, μ = exp(−3x).