Question

(1 point) The equation z2 can be written in the form y'-f(y/z), ie., it is homogeneous, so we can use the substitution u-y/z to obtain a separable equation with dependent variable Introducing this substitution and using the fact that y' zuu we can write (*) as u u- f(u) where f(u) Separating variables we can write the equation in the form dz where g(u)- An implicit general solution with dependent variable u can be written in the form In(z) Transforming u y/z back into the variables z and y and using the initial condition y(1) 3 we find Finally solve for y to obtain the explicit solution of the initial value problem

Answer 1

u = y/x y = ux y' = u + xu' 4y' = 4u + 4xu' = 4 (y/x) + 9 (y/x)^2 4u + 4xu' = 4u + 9 + u^2 4xu' = 9 + u^2 xu' = (9 + u^2)/4 f (u) = (9 + u^2)/4 4/9 + u^2 du = dx/x g (u) = 4/9 + u^2) Integrating gives 4/3 tan^-1 (u/3) + C = ln (x) ln (x) - 4/3 tan^-1 (u/3) = C ln (x) - 4/3 tan^-1 (y/3x) = C y (1) = 3 ln(1) - 4/3 tan^-1 (3/3 * 1) = C - 4/3 tan^-1 (1) = C = -pi/3 3 ln (x) /4 + pi/4 = tan^-1 (y/3x) y/(3x) = tan(3 ln (x)/4 + pi/4) y = 3x tan (3 ln (x)/4 + pi/4)