Write the equation in the form y′=f(y/x) then use the substitution y=xu to find an implicit general solution. Then solve the initial value problem.

y′=5y2+4x2xy, y(1)=3

The resulting differential equation in x x and u u can be written as xu′= which is separable.

Separating variables we arrive at du=dx/x

Separating variables and and simplifying the solution can be written in the form u2+1=Cf(x) where C is an arbitrary constant and f(x)= .

Transforming back into the variables x and y and using the initial condition to find C we find the explicit solution of the initial value problem is y = .

Ahigh-altitude spherical weather balloon expands as it rises, due to the drop in atmospheric pressure. suppose that the radius r increases at the rate of 0.02 inches per second, and that r = 36 inches at time t = 0. determine the equation that models the volume v of the balloon at time t, and find the volume when t = 360 seconds. v(t) = 4π(0.02t)2; 651.44 in3 v(t) = 4π(36 + 0.02t)2; 1,694,397.14 in3 v(t) = four pi times the product of zero point zero two and t to the third power divided by three.; 4,690.37 in3 v(t) = four pi times the quantity of thirty six plus zero point zero two t to the third power divided by three.; 337,706.83 in3