Let f(x)=x−1 and g(x)=ex for x∈R. If dxdy=(e−2xg(f(f(f(x))))−xy),y(0)=0, then y(1) is:
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Step-by-Step Solution
Step 1: Simplify the composite function
First, we need to simplify the composite function f(f(f(x))). We apply f(x) three times.
f(x)=x−1 f(f(x))=f(x−1)=(x−1)−1=x−2 f(f(f(x)))=f(x−2)=(x−2)−1=x−3.
Step 2: Substitute into g(x) and the differential equation
Now we substitute f(f(f(x)))=x−3 into g(x)=ex, which gives us g(f(f(f(x))))=ex−3. Then, we substitute this back into the given differential equation.
Step 3: Rearrange into a linear differential equation
We rearrange the differential equation into the standard form of a linear first-order differential equation, which is dxdy+P(x)y=Q(x). Here, P(x)=x1 and Q(x)=ex−3−2x.
Step 4: Find the integrating factor
The integrating factor (I.F.) for a linear differential equation is given by e∫P(x)dx. In this case, ∫x1dx=∫x−1/2dx=2x1/2=2x. So, the integrating factor is e2x.
Step 5: Solve the differential equation
Multiply both sides of the differential equation by the integrating factor. The left side becomes dxd(y⋅I.F.). The right side simplifies to ∫ex−3dx=ex−3+C.
Step 6: Apply the initial condition
We use the initial condition y(0)=0 to find the value of the constant C. Substituting x=0 and y=0 into the solution gives 0=e−3+C, so C=−e−3.
Step 7: Find y(1)
Substitute the value of C back into the general solution to get the particular solution. Then, substitute x=1 to find y(1).