Answer:
y(x) = -x sqrt(c_1 x - 1) or y(x) = x sqrt(c_1 x - 1)
Step-by-step explanation:
Solve Bernoulli's equation ( dy(x))/( dx) = (x^2 + 3 y(x)^2)/(2 x y(x)):
Rewrite the equation:
( dy(x))/( dx) - (3 y(x))/(2 x) = x/(2 y(x))
Multiply both sides by 2 y(x):
2 ( dy(x))/( dx) y(x) - (3 y(x)^2)/x = x
Let v(x) = y(x)^2, which gives ( dv(x))/( dx) = 2 y(x) ( dy(x))/( dx):
( dv(x))/( dx) - (3 v(x))/x = x
Let μ(x) = e^( integral-3/x dx) = 1/x^3.
Multiply both sides by μ(x):
(( dv(x))/( dx))/x^3 - (3 v(x))/x^4 = 1/x^2
Substitute -3/x^4 = d/( dx)(1/x^3):
(( dv(x))/( dx))/x^3 + d/( dx)(1/x^3) v(x) = 1/x^2
Apply the reverse product rule f ( dg)/( dx) + g ( df)/( dx) = d/( dx)(f g) to the left-hand side:
d/( dx)(v(x)/x^3) = 1/x^2
Integrate both sides with respect to x:
integral d/( dx)(v(x)/x^3) dx = integral1/x^2 dx
Evaluate the integrals:
v(x)/x^3 = -1/x + c_1, where c_1 is an arbitrary constant.
Divide both sides by μ(x) = 1/x^3:
v(x) = x^2 (c_1 x - 1)
Solve for y(x) in v(x) = y(x)^2:
Answer: y(x) = -x sqrt(c_1 x - 1) or y(x) = x sqrt(c_1 x - 1)
