Final answer:
The integrating factor for the given linear differential equation is e^(x^2). The new differential equation after multiplying both sides by the integrating factor is d/dx (e^(x^2) * y) = x^7 * e^(-x^2/2). After integrating both sides and using the integration constant, the algebraic equation becomes e^(x^2) * y = ∫(x^7 * e^(-x^2/2) dx) + C. The solution to the differential equation is y = (1/e^(x^2)) * ∫(x^7 * e^(-x^2/2) dx) + C/e^(x^2). The solution of the initial value problem (IVP) with the condition y(0) = 18 is y = (1/e^(x^2)) * ∫(x^7 * e^(-x^2/2) dx) + C/e^(x^2).
Step-by-step explanation:
a) To find the integrating factor for the given linear differential equation, we need to look at the coefficient of y, which is 2x. The integrating factor is e^(∫2x dx), which simplifies to e^(x^2).
b) After multiplying both sides of the differential equation by the integrating factor, the new differential equation is d/dx (e^(x^2) * y) = x^7 * e^(-x^2/2).
c) Integrating both sides of the new differential equation, we get e^(x^2) * y = ∫(x^7 * e^(-x^2/2) dx). The algebraic equation is e^(x^2) * y = ∫(x^7 * e^(-x^2/2) dx) + C, where C is the integration constant.
d) Solving for y, the solution to the differential equation is y = (1/e^(x^2)) * ∫(x^7 * e^(-x^2/2) dx) + C/e^(x^2).
e) To find the solution of the initial value problem (IVP) with the initial condition y(0) = 18, substitute the value of x and y into the solution equation to solve for C. The solution of the IVP is y = (1/e^(x^2)) * ∫(x^7 * e^(-x^2/2) dx) + C/e^(x^2).