Solve the following equations by variation of parameters If

Solve the following equations by variation of parameters If y1(x) = x2 is a solution of x2 y\" -3xy +4y = 0, find the general solution of this equation on the interval y\" +y = see theta tan theta

Solution

b.)y(x) = c_2 sin(x)+c_1 cos(x)+x cos(x)-sin(x) log(cos(x))
a.)Solve the separable equation x^2 ( dy(x))/( dx)-3 x ( dy(x))/( dx)+4 y(x) = 0:
Solve for ( dy(x))/( dx):
( dy(x))/( dx) = -(4 y(x))/(x (x-3))
Divide both sides by y(x):
(( dy(x))/( dx))/(y(x)) = -4/(x (x-3))
Integrate both sides with respect to x:
integral (( dy(x))/( dx))/(y(x)) dx = integral -4/(x (x-3)) dx
Evaluate the integrals:
log(y(x)) = -4 (-(log(x))/3+1/3 log(-x+3))+c_1, where c_1 is an arbitrary constant.
Solve for y(x):
y(x) = (e^(c_1) x^(4/3))/(-x+3)^(4/3)
Simplify the arbitrary constant:
y(x) = (c_1 x^(4/3))/(-x+3)^(4/3)