Consider the differential equation x²d²y + xdy

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Consider the differential equation x² d²y + x dy - 4y = 0
dx² dx

with the boundary conditions of y(0) with the boundary conditions of y(0) = 0 and y(1) = 1. The complete solution of the differential equation is

1. x²
2. sin πx
  2
3. e^xsin πx
  2
4. e^-xsin πx
  2

Correct Option: A

x²	d²y	+ x	dy	- 4y = 0
	dx²		dx

⇒ x²y" + xy' - xy = 0
From Euler-Cauchy’s Equation a₀ x²y" + a₁ + xy' + a₂y = 0
Given: a₀ = 1, a₁ =1, a₂ = – 4
∴ a₀m(m– 1) + a₁m + a₂ = 0
⇒ m(m – 1) + m – 4 = 0
⇒ m₂ = 4
⇒ m = ± 2

y(x) = Ax² + Bx^–2 = Ax² +	B
	x²

Applying Boundary condition

y(0) = A0 +	B	= 0
	0

∴ B = 0
and y(1) = A(1) = 1
∴ A = 1
y(x) = x²

Consider the differential equation x²	d²y	+ x	dy	- 4y = 0
	dx²		dx

sin		πx
		2

e^xsin		πx
		2

e^-xsin		πx
		2