Final answer:
To find the solution of the given initial value problem, we need to solve the non-homogeneous differential equation y' - y = 2te^2t. Following the steps, we first find the general solution to the homogeneous equation and then a particular solution to the non-homogeneous equation. By combining the two, we obtain the general solution to the non-homogeneous equation. Finally, we use the initial condition to find the constant and write the final solution.
Step-by-step explanation:
Solution:
Step 1:
Identify the homogeneous equation: y' - y = 0
Step 2:
Solve the homogeneous equation by finding the general solution.
The general solution to the homogeneous equation is y
h
= Ce
t
where C is a constant.
Step 3:
Find a particular solution to the non-homogeneous equation.
We can use the method of undetermined coefficients to find a particular solution.
Assume a particular solution of the form y
p
= At
2
e
t
where A is a constant.
Differentiate y
p
to find y'
p
and substitute into the non-homogeneous equation:
y'
p
- y
p
= 2te
t
e
t
- At
2
e
t
= 2te
2t
-At
2
e
t
Equating like terms, we get 2te
2t
- At
2
e
t
= 2te
t
e
t
By comparing coefficients, we find that A = -1/2
Therefore, a particular solution to the non-homogeneous equation is y
p
= (-1/2)t
2
e
t
Step 4:
Find the general solution to the non-homogeneous equation.
The general solution to the non-homogeneous equation is y = y
h
+ y
p
.
Substituting the values for y
h
and y
p
we obtained earlier, we have:
y = Ce
t
+ (-1/2)t
2
e
t
Step 5:
Use the initial condition y(0) = 1 to find the constant C.
Substituting 0 for t and 1 for y in the general solution:
1 = Ce
0
+ (-1/2)(0)
2
e
0
C = 1
Step 6:
Write the final solution
The final solution to the given initial value problem is y = e
t
- (1/2)t
2
e
t