Let y=f(x) be the solution of the differential equation dxdy+x2−1xy=1−x2x4+4x,−1<x<1 such that f(0)=0. If 6∫−1/21/2f(x)dx=2π−α then α2 is equal to:
Get the complete, step-by-step math solution for: "Let y=f(x) be the solution of the differential equation (dy)/(dx) + (xy)/(x²-1) = {x^4 + 4x}{√(1-x²)}, -1 < x < 1 such that f(0)=0. If 6 ∫_(-1/2)^(1/2...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the form of the differential equation and find the integrating factor
The given differential equation is a first-order linear differential equation of the form dxdy+P(x)y=Q(x). We first identify P(x) and then calculate the integrating factor (I.F.) using the formula e∫P(x)dx. The integral of x2−1x can be solved by substitution, letting u=x2−1, so du=2xdx.
Step 2: Calculate the integrating factor
We evaluate the integral for P(x). By substituting u=x2−1, the integral becomes 21ln∣x2−1∣. Since −1<x<1, x2−1 is negative, so ∣x2−1∣=−(x2−1)=1−x2. Thus, the integrating factor is eln1−x2, which simplifies to 1−x2.
Step 3: Solve the differential equation
Now we use the formula for the solution of a linear differential equation: y⋅I.F.=∫Q(x)⋅I.F.dx. We substitute Q(x) and the integrating factor into the equation. The 1−x2 terms cancel out, simplifying the integral. We then integrate x4+4x with respect to x to find the general solution.
Step 4: Apply the initial condition to find C
We are given the initial condition f(0)=0. We substitute x=0 and y=0 into the general solution to find the value of the constant C. This substitution directly yields C=0.
Step 5: Determine f(x) and evaluate the integral
With C=0, the solution to the differential equation is y=f(x)=51−x2x5+10x2. We need to evaluate the definite integral 6∫−1/21/2f(x)dx. We can split the integral into two parts: one for the odd function x5 and one for the even function x2.
Step 6: Simplify the integral using properties of odd and even functions
The term 51−x2x5 is an odd function, so its integral over a symmetric interval [-a, a] is 0. The term 51−x210x2=1−x22x2 is an even function. For an even function, ∫−aag(x)dx=2∫0ag(x)dx. This simplifies the integral to 24∫01/21−x2x2dx.
Step 7: Evaluate the remaining integral and find α2
To evaluate ∫1−x2x2dx, we use the trigonometric substitution x=sinθ. This transforms the integral into ∫sin2θdθ, which can be solved using the identity sin2θ=21−cos(2θ). After integrating and substituting back x, we evaluate the definite integral from 0 to 1/2. The result is 2π−33. Comparing this to 2π−α, we find α=33. Finally, we calculate α2.