Math Solutions

Let U =(1, 2, 3, 4, 5, 6, 7, 8, 91, A =11, 2, 3, 41, B=(2, 4, 6, 8 ) and C=13, 4, 5, 61. Find (i) A’ (ii) B’ (ii) (AuC)' (iv) (A U B)' (v)(A'(vi) (B- C)

A cup of coffee at 90°C cools in a 20°C room. Assuming dT/dt = −k·(T − T_amb), solve for T(t) and plot the cooling curve. Slide k to see how the rate changes.

Approximate the solution of y′ = x + y with y(0) = 1 on 0 ≤ x ≤ 1 using Euler's method with step size h = 0.2. Compare against the exact solution y = 2e^x − x − 1.

Show that the IVP y′ = √y with y(0) = 0 has multiple solutions: y ≡ 0 and y = t²/4. Explain why Picard's uniqueness theorem does not apply here.

A series RLC circuit with R = 10 Ω, L = 1 H, C = 0.01 F carries charge Q(t). Solve L·Q″ + R·Q′ + Q/C = 0 and show the damped oscillation of charge and current.

A 2 kg mass is attached to a spring with stiffness k = 50 N/m and damping c = 4 N·s/m. Solve m·x″ + c·x′ + k·x = 0 and show the decaying oscillation.

A 100-litre tank initially contains pure water. Brine at 5 g/L flows in at 2 L/min; the well-stirred mixture drains at 2 L/min. Find the salt amount S(t) at time t.

If y1 and y2 both solve y'' + y = 0, show that c1 y1 + c2 y2 is also a solution. Animate the sum of cos x and sin x combining into a new oscillation.

Given that y1 = e^x solves y'' - 2 y' + y = 0, find a second linearly independent solution using reduction of order. Show that y2 = x e^x emerges naturally.

Solve y'' + y = 0 with y(0) = 0, y(pi) = 0. Identify the eigenvalue condition that produces non-trivial solutions and explain why infinitely many exist.

For x' = A x with A = [[0, 1], [-2, -3]], find the eigenvalues and classify the equilibrium at the origin (stable node, saddle, spiral, or centre). Draw representative trajectories.

Solve dx/dt = x + y, dy/dt = -x + y. Find the eigenvalues of the system matrix, recognise a spiral, and show the outward-spiraling solution in the phase plane.

Compute W(e^x, e^(2x)) and show it equals e^(3x), which is non-zero, confirming the two functions are linearly independent. Display the Wronskian determinant.

Solve y' = y using a power series y = sum of a_n x^n, n = 0 to infinity. Show the recursion for a_n and recover the exponential y = e^x term by term.

Compute (f * g)(t) = integral from 0 to t of f(tau) g(t - tau) d tau for f(t) = e^(-t) and g(t) = t. Verify the Laplace convolution theorem L{f * g} = F(s) G(s), and animate the sliding integral.

Express f(t) = 0 for t < 2 and 5 for t >= 2 using the Heaviside step u(t - 2). Find its Laplace transform, and show the second shifting theorem in action.

Solve y'' + 4y = delta(t - 2) with y(0) = 0 and y'(0) = 0. Show the impulse response and the delayed-sine waveform that emerges from the Dirac delta forcing.

Find L^{-1}{(s+1) / (s^2 + 2s + 5)}. Complete the square in the denominator, apply the shifting theorem in reverse, and present the time-domain result.

Find L{e^(2t) sin(3t)}. Use the shifting theorem and the standard transform of sin(bt). Show the s-domain representation on a pole-zero diagram.

Solve x^2 y'' - 2 x y' + 2 y = 0. Substitute y = x^r to reduce to an algebraic equation in r and show the resulting power-law solutions.