Let y = Φ(x) and y = Ψ(x) be solutions of y" - 2xy + (sin x2)y = 0 such that Φ(0) = 1, Φ(0) =1 andΨ(0) =1, Ψ0)=2. Then the value of Wronskian w(Φ,Ψ) at x=1 is (upto two decimal places) __________.Correct answer is '2.71'. Can you explain this answer?

Question

Let y = &Phi;(x) and y = &Psi;(x) be solutions of y" - 2xy + (sin x2)y = 0 such that &Phi;(0) = 1, &Phi;(0) =1 and&Psi;(0) =1, &Psi;0)=2. Then the value of Wronskian w(&Phi;,&Psi;) at x=1 is (upto two decimal places) __________.Correct answer is '2.71'. Can you explain this answer?

Answer

y" - 2xy' + (sin x²)y = 0 ........(i)

Compare with y" + P(x)y' +Q(x)y= 0

we have P(x) =-2x, Q(x) = sin x²

∴ w(Φ, Ψ) = Φ(x) Ψ'(x) - Φ'(x) Ψ(x)

⇒ w(x) = c e^x² = Φ(x) ·Ψ'(x) - Φ'(x) ·Ψ(x)

∴ w(x) = c = Φ(2)· Ψ'(0) - Φ'(0) · Ψ(0)

= 1 . 2 - 1 . 1

= 2 - 1

= 1

⇒ w(x) = e^x²

⇒ w(1) = e = 2.71

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