For x∈R, x≠0, if y(x) is a differentiable function such that ∫₁ˣ y(t)dt = (x+1)∫₁ˣ ty(t)dt, then y(x) equals: (Where C is a constant) Cx³e^(1/x), Cxe^(1/x), Cx³e¹ˣ, Cx²e^(1/x)

∫₁ˣy(t)dt = x∫₁ˣty(t)dt + ∫₁ˣty(t)dt
differentiate w.r. to x.
∫₁ˣy(t)dt + x[y(x) - y(1)] = ∫₁ˣty(t)dt + x[xy(x) - y(1)] + xy(x) - y(1)
∫₁ˣy(t)dt = ∫₁ˣty(t)dt + x²y(x) - y(1)
diff. again w.r to x
y(x) - y(1) = xy(x) - y(1) + 2xy(x) + x²y'(x)
(1-x)y(x) = x²y'(x)
y'(x)/y(x) = (1-x)/x²
1/y dy/dx = 1/x² - 1/x
∫1/y dy = ∫(1/x² - 1/x)dx
ln y = -1/x - ln x + C
ln y = -1/x - ln x + ln c
ln y = ln(ce⁻ˣ⁻¹/x)
y = ce⁻ˣ⁻¹/x
ln(y x³) = -1/x + ln c
yx³ = ce⁻ˣ⁻¹
y = ce⁻ˣ⁻¹/x³
or y = cx³e⁻ˣ⁻¹