Explanation
Solution
1. Standard Form of the Differential Equation
First, we divide the entire equation by (x−1) to put it into the standard linear form dxdy+P(x)y=Q(x):
Here, P(x)=x−12x and Q(x)=(x−1)21.
2. Finding the Integrating Factor (I.F.)
The integrating factor is given by e∫P(x)dx:
∫x−12xdx=∫x−12(x−1)+2dx=∫(2+x−12)dx=2x+2ln(x−1)
I.F.=e2x+2ln(x−1)=e2x⋅eln(x−1)2=(x−1)2e2x
3. General Solution
The solution is given by y⋅(I.F.)=∫Q(x)⋅(I.F.)dx+C:
y(x−1)2e2x=∫(x−1)21(x−1)2e2xdx+C
4. Applying Initial Condition y(2)=2e41+e4
Substitute x=2:
(2e41+e4)(2−1)2e2(2)=2e2(2)+C
So, the particular solution is:
5. Finding y(3)
Substitute x=3 into the particular solution:
y(3)=2(3−1)2e2(3)e2(3)+1=2(4)e6e6+1=8e6e6+1
6. Comparing with the given form
The given form is y(3)=βeαeα+1.
By comparison:
7. Final Value
The value of α+β is 14.
