Sometimes a change of variable can be used to convert a differential equation y′=f(t, y) into a separable equation. one common change of variable technique is as follows. consider a differential equation of the form y′=f(αt+βy+γ), where α,β, and γ are constants. use the change of variable z=αt+βy+γ to rewrite the differential equation as a separable equation of the form z′=g(z). solve the initial value problem y′=(t+y)2−1, y(3)=4.

Step-by-step explanation:

We are given that

Initial value problem

, y(3)=4

Substitute the value

When t=3 and y=4 then

z=3+4=7

Differentiate z w.r.t t

Then, we get

Integrate on both sides

Substitute t=3 and z=7

Then, we get

Substitute the value of C then we get

80π/3 or about 83.775

step-by-step explanation:

since a cone is 1/3 of a cylinder, we need to find the base's area. since area = πr^2, we get 16π. then, multiplying this by 5, we get 80π. dividing by 3, since the cone's volume is 1/3 of that of a cylinder's, we get 80π/3.

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