Final answer:
To solve the given differential equation, we can use the method of integrating factors. We rewrite the equation in the form y' + P(x)y = Q(x) and find the integrating factor. Then, we multiply the equation by the integrating factor, integrate both sides, and solve for the constant of integration. Finally, we substitute the given values and solve for y when x = 1/2.
Step-by-step explanation:
To solve the differential equation, we can use the method of integrating factors. First, we rewrite the equation in the form y' + P(x)y = Q(x), where P(x) = 1/x and Q(x) = x^2/x. Next, we find the integrating factor, which is given by e^(∫P(x)dx) = e^(∫(1/x)dx) = e^(ln|x|) = |x|. We multiply the entire equation by the integrating factor |x| and then integrate both sides with respect to x. The resulting equation is ∫y/x dx + ∫(x^2/x)|x| dx = ∫|x| dx.
Integrating the left side, we get ∫y/x dx + ∫|x| dx = |x|^2/2 + c, where c is the constant of integration. Substitute in y = y = y(x) and integrate with respect to x to get ∫y/x dx = ∫|x|^2/2 + c - ∫|x| dx. Simplifying, we have y(x) = ((|x|^2 + 2c)/2)ln|x| + c'ln|x| + c'', where c' and c'' are constants of integration.
To find the specific solution to the initial value problem, we substitute the given values y(2) = 3 and x = 2 into the equation. This gives us 3 = ((|2|^2 + 2c)/2)ln|2| + c'ln|2| + c''. Simplifying and solving for c gives c = (3 - 2c'ln|2| - c'')/(1 + ln(4)).
Finally, we can substitute x = 1/2 into the equation y(x) = ((|x|^2 + 2c)/2)ln|x| + c'ln|x| + c'' and solve for y. This will give us the value of y when x = 1/2.