It is given that y = eˣu, where u is a function of x. The rth derivatives dʳy/dxʳ and dʳu/dxʳ are denoted by y⁽ʳ⁾ and u⁽ʳ⁾ respectively. Prove by mathematical induction that, for all positive integers n, y⁽ⁿ⁾ = eˣ ((ⁿ₀) u + (ⁿ₁) u⁽¹⁾ + (ⁿ₂) u⁽²⁾ + ... + (ⁿᵣ) u⁽ʳ⁾ + ... + (ⁿₙ) u⁽ⁿ⁾). [You may use without proof the result (ᵏᵣ) + (ᵏᵣ₋₁) = (ᵏ⁺¹ᵣ)].

Given that a is a constant, prove by mathematical induction that, for every positive integer n, dⁿ/dxⁿ(xeᵃˣ) = naⁿ⁻¹e…

Using factorials, show that (n/(r-1)) + (n/r) = ((n+1)/r). Hence prove by mathematical induction that (a + x)ⁿ = (n/0…

Show that dⁿ⁺¹/dxⁿ⁺¹ (xⁿ⁺¹ lnx) = dⁿ/dxⁿ (xⁿ + (n + 1)xⁿ lnx).

Find the general solution of the differential equation d²x/dt² + 2dx/dt + x = 4 sin t.

The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = 3 2 0 1 6 5 -1 3 9 8 -2 …

Let S_N = Σ(from r=1 to N) (3r+1)(3r+4) and T_N = Σ(from r=1 to N) 1/((3r+1)(3r+4)).