Math Solutions

Solve y'' + y = sec(x) using variation of parameters. Show why undetermined coefficients cannot handle sec(x), then derive the particular solution from the Wronskian.

Solve y'' + y = sin(2x). Guess y_p = A sin(2x) + B cos(2x), determine A and B, then combine with the homogeneous solution.

Solve y'' - 6y' + 9y = 0. Characteristic equation (r-3)^2 = 0 gives repeated root r = 3. Write the general solution y = (C1 + C2 x) e^(3x) and show why the x e^(3x) term is needed.

Solve y'' + 4y = 0. Find characteristic roots r = plus or minus 2i and write the real-valued general solution y = C1 cos(2x) + C2 sin(2x). Show the oscillation and its frequency.

Solve y'' - y = 0. Find characteristic roots r = plus or minus 1 and write the general solution y = C1 e^x + C2 e^(-x). Show both exponential modes and their hyperbolic-function recombination.

For y'' + 4y' + 4y = 0, form the characteristic equation r^2 + 4r + 4 = 0. Factor it, find r = -2 (repeated), and write the general solution y = (C1 + C2 x) e^(-2x).

Solve y'' - 5y' + 6y = 0. Find the characteristic roots r = 2, 3 and write the general solution y = C1 e^(2x) + C2 e^(3x).

Solve dy/dx = 2x with y(0) = 3. Show the unique solution y = x^2 + 3 and explain how the initial condition picks it out of the family y = x^2 + C.

Plot the slope field of dy/dx = x - y, then overlay several solution curves that follow the field. Show how curves trace the flow.

Solve dy/dx = (x^2 + y^2) / (x y). Substitute y = v x to reduce the equation to separable form, and show the substitution in action.

Solve dy/dx + y = y^3 using the substitution v = y^(-2). Show how the substitution reduces a non-linear ODE to a linear one, then reconstruct the solution curves.

Verify that (2xy + 3) dx + (x^2 + 4y) dy = 0 is exact, then find the potential function F(x, y) so that dF = 0 along every solution.

Solve dy/dx + (1/x) y = x using the integrating factor mu(x) = x. Animate the multiplication step that collapses the left side into a single derivative.

Solve dy/dx + 2y = 6 and show the solution approaching the equilibrium y = 3.

Solve dy/dx = xy. Separate variables to get dy/y = x dx, integrate both sides, and show the family of solution curves.

Diagonalize the symmetric matrix A = [[2,1],[1,2]] with orthogonal eigenvectors. Show the eigendirections are perpendicular.

Show that A = [[2,1],[1,3]] is positive definite. Verify by computing eigenvalues, checking the quadratic form, and visualizing the ellipsoidal level curves.

Express v = [5, 3] in the basis B = {[1,1], [1,−1]}. Find the new coordinates and show the relationship between the two coordinate grids.

Decompose A = [[3,2],[2,3]] as A = UΣVᵀ. Show how the singular values reveal the geometry of the transformation.

Compute the inner product ⟨f, g⟩ = ∫₀¹ f(x)g(x) dx for f(x) = x and g(x) = x². Interpret the result geometrically.