← Differential Equations

Unit 6: Second-Order Linear Homogeneous Equations

Guessing y = e^(rt) turns calculus into algebra: the characteristic equation, its three root cases, and how the Wronskian proves two solutions are enough.

10 topics · 34 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

3.1 Second-Order Linear Equations and Homogeneity

  1. How it worksHow do you tell a homogeneous second-order linear equation apart from a nonhomogeneous one just by looking at it?
  2. Common mistakeWhy isn't a homogeneous linear equation the same idea as an equation whose functions are homogeneous of some degree?
  3. Going furtherWhat happens to the number of independent solutions once the equation's order climbs from two to three?
  4. Worked exampleFind whether y'' + 4y' - 2y = 0 is homogeneous or nonhomogeneous, and state its order.
Your turn1 of 2
Classify the equation 2y'' + 2y' - 3y = 2: is it homogeneous or nonhomogeneous, and what order is it?

3.2 The Principle of Superposition

  1. How it worksHow does linearity make any combination c1 y1 plus c2 y2 satisfy the same homogeneous equation y1 and y2 do?
  2. Worked exampleShow that 3cos(2x) - 5sin(2x) solves y'' + 4y = 0, given that y1 = cos(2x) and y2 = sin(2x) both solve it.
  3. Common mistakeWhy doesn't superposition work the same way once the equation has a nonzero forcing term on the right?
Your turn1 of 2
Given that y1 = cos(2x) and y2 = sin(2x) both solve y'' + 22 y = 0, find the value of y(x) = 4y1(x) + 5y2(x) at x = 0.551.

3.1 The Characteristic Equation

  1. How it worksHow does substituting y = ert into ay'' + by' + cy = 0 produce the characteristic equation ar2 + br + c = 0?
  2. Worked exampleFind the characteristic equation and its roots for y'' - 5y' + 6y = 0.
  3. Going furtherWhat happens to the characteristic equation when the differential equation has a third derivative in it?
  4. Common mistakeWhy doesn't a characteristic equation with no real roots mean the differential equation has no real solutions?
Your turn1 of 2
Find the larger root of the characteristic equation r2 - 9r + 4 = 0, coming from y'' - 9y' + 4y = 0. (Both roots are real.)

3.1 Real and Distinct Roots

  1. How it worksHow do er1 t and er2 t combine into the general solution when the roots r1 and r2 are real and distinct?
  2. Worked exampleSolve y'' - y' - 6y = 0 given the initial conditions y(0) = 1 and y'(0) = 0.
  3. Common mistakeWhy isn't the general solution just er1 t plus er2 t with no constants multiplying them?
Your turn1 of 2
Solve y'' - (5 - 3)y' - 5·3y = 0 with y(0) = 5 and y'(0) = -1. (Its characteristic roots are 5 and -3.) Find y(t).

3.3 Complex Roots and Euler's Formula

  1. How it worksHow does Euler's formula turn e to the (a+bi)t power into a real oscillation multiplied by a real exponential envelope?
  2. How it worksHow do you go from a complex conjugate pair of roots to the real solution eat times C1 cos(bt) plus C2 sin(bt)?
  3. Worked exampleSolve y'' + 4y' + 13y = 0 given that its characteristic equation has complex roots.
  4. Common mistakeWhy doesn't a negative number under the square root in the quadratic formula mean something has gone wrong?
Your turn1 of 2
Solve y'' + 4y' + 24y = 0 with y(0) = 4 and y'(0) = -1. Find y(t). (Its characteristic equation has complex roots.)

3.3 Solving an Oscillating Second-Order Equation

  1. Worked exampleSolve y'' + 9y = 0 with y(0) = 2 and y'(0) = 0.
  2. Worked exampleFind the amplitude and period of the solution to y'' + 4y' + 20y = 0 with y(0) = 1, y'(0) = 0.
  3. Going furtherWhat happens to the oscillation's envelope when the real part of the complex roots is exactly zero?
Your turn1 of 2
Solve y'' + 42 y = 0 with y(0) = 3 and y'(0) = 3. Find y(t).

3.4 Repeated Roots and the Second Solution

  1. How it worksHow does multiplying by t turn ert into a genuinely new second solution when the root is repeated?
  2. Worked exampleSolve y'' - 6y' + 9y = 0 with y(0) = 0 and y'(0) = 2.
  3. Common mistakeWhy isn't t2 ert also a valid second solution alongside t ert for a repeated root?
Your turn1 of 2
Solve y'' - (3·2)y' + 32 y = 0 with y(0) = 2 and y'(0) = 4, given that r = 3 is a repeated root. Find y(t).

3.2 The Wronskian and Linear Independence

  1. How it worksHow does the determinant of y1, y2 and their derivatives detect whether one solution is a multiple of the other?
  2. Worked exampleFind the Wronskian of y1 = e2x and y2 = e−3x and use it to confirm they're independent.
  3. Going furtherWhat happens to the Wronskian of two solutions of a homogeneous linear equation as x ranges over all values?
  4. Worked exampleFind the Wronskian of y1 = t and y2 = t ln(t) and use it to show they're independent for t > 0.
Your turn1 of 2
Find the Wronskian W(x) of y1 = e3x and y2 = e−3x.

3.2 When Two Solutions Aren't a Fundamental Set

  1. How it worksHow does a zero Wronskian reveal that two solutions are secretly proportional to each other?
  2. Worked exampleShow that y1 = e3x and y2 = 4e3x both solve y'' - 6y' + 9y = 0 but their Wronskian is zero.
  3. Common mistakeWhy doesn't finding two different functions that both satisfy an equation guarantee you have its general solution?
Your turn1 of 2
Given that y1 = 3e4x and y2 = 8e4x both solve the same homogeneous equation, find the ratio y2(x)/y1(x). (Notice it doesn't depend on x at all.)

3.4 Reduction of Order

  1. How it worksHow does substituting y = v(t) times y1(t) turn a second-order equation for y into a first-order equation for v prime?
  2. Worked exampleFind a second, independent solution of t2 y'' + t y' - y = 0 using reduction of order, given that y1 = t solves it.
  3. Going furtherWhat happens to the reduction of order method when the known solution y1 happens to be zero somewhere on the interval?
Your turn1 of 2
Given that y1 = t solves t2y'' + 4t y' - 4y = 0, use reduction of order to find a second, independent solution y2(t).