**Answer:**

**Step-by-step explanation:**

**Solution:-**

- A change of variable is a technique employed in solving many differential equations that are of the form: y ' = f ( t , y ).

- Considering a differential equation of the form y' = f ( αt + βy + γ ), where α, β, and γ are constants. A substitution of an arbitrary variable z = αt + βy + γ is made and the given differential equation is converted into a form: z ' = g ( z ).

- This substitution basically allow us to solve in-separable differential equations by converting them into a form that can be separated, followed by the set procedure.

- We are to solve the initial value problem for the following differential equation:

**First Step: Make the appropriate substitution**

- We will use a arbitrary variable ( z ) and define the our substitution by finding a multi-variable function f ( t , y ) that is a part of the given ODE.

- We see that the term ( t + y ) is a multi-variable function and also the culprit that doesn't allow us to separate our variables.

- Usually, the change of variable substitution is made for such " **culprits** ".

- So our substitution would be:

**Second Step:** Implicit differential of the substitution variable ( z ) with respect to the independent variable

- In the given ODE we see that the variable ( t ) is our independent variable. So we will derivate the supposed substitution as follows:

**Remember:** z is a multivariable function of "t" and "y". So we perform implicit differential for the variable " z ".

**Third Step:** Plug in the differential form in step 2 and change of variable substitution of ( z ) in the given ODE.

- The given ODE can be expressed as:

... **Separable ODE**

**Fourth Step: **Separate the variables and solve the ODE.

- We see that the substitution left us with a simple separable ODE.

**Note:** If we **do not** arrive at a **separable ODE**, then we must go back and re-choose our change of variable substitution for ( z ).

- We will progress by solving our ODE:

Where,

c: The constant of integration

**Fifth Step: **Back-substitution of variable ( z )

- We will now back-substitute the substitution made in the first step and arrive back at our original variables ( y and t ) as follows:

**Sixth Step:** Apply the initial value problem and solve for the constant of integration ( c )

- We will use the given initial value statement i.e y ( 3 ) = 6 and evaluate the constant of integration ( c ) as follows:

**Seventh Step: **Express the solution of the ODE in an explicit form ( if possible ):