Final answer:
The differential equation dy/dx - y - e³⁸ = 0 is solved using the method of integrating factors to find the general solution y = -0.5e²⁸ + Ce⁸, where C is the integration constant.
Step-by-step explanation:
Solving the Differential Equation
To solve the differential equation dy/dx - y - e³⁸ = 0, we will use the method of integrating factors. Firstly, we identify the equation as a first-order linear non-homogeneous differential equation. The standard form is dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x. In our case, P(x)=-1 and Q(x) is e³⁸.
The integrating factor, μ(x), is found using the formula μ(x) = e∫P(x)dx, which in this case yields μ(x) = e-x. Multiplying through the differential equation by this integrating factor will allow us to rewrite the left side as a product derivative of y and μ(x). This leads us to:
e-xdy/dx - e-xy = e2x
The left side becomes the derivative d/dx of (y e-x). We integrate both sides with respect to x to obtain y e-x = ∫ e2x dx + C, where C is the integration constant. After integrating the right side and simplifying, we find y as:
y = ex (∫ e2x dx + C)
To solve the integral, we find y = -0.5e2x + Cex, which is the general solution to the differential equation.