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

what

1) the property is   ixi = a ⇔x = a   o x = -a

4(x-5)+3= 15⇒4x-20+3 = 15⇒4x-17= 15⇒4x = 15+17⇒4x = 32⇒x =32/4= 8

4(x-5) +3= -15⇒4x-20 +3 = -15⇒4x-17= -15⇒4x =-15+17⇒4x = 2⇒x =2/4= 1/2

the values are x =8 and x = 0,5