Answer:
y(x) = c_1 e^(-1/(2 x^2))
Explanation:
Solve the separable equation x^3 (dy(x))/(dx) - y(x) = 0:
Solve for (dy(x))/(dx):
(dy(x))/(dx) = y(x)/x^3
Divide both sides by y(x):
((dy(x))/(dx))/y(x) = 1/x^3
Integrate both sides with respect to x:
integral((dy(x))/(dx))/y(x) dx = integral1/x^3 dx
Evaluate the integrals:
log(y(x)) = -1/(2 x^2) + c_1, where c_1 is an arbitrary constant.
Solve for y(x):
y(x) = e^(-1/(2 x^2) + c_1)
Simplify the arbitrary constants:
Answer: y(x) = c_1 e^(-1/(2 x^2))