Differential equations

Worked examples (6)

• 1Rearranging to standard form and reading off P and Q
• 2Constant P: the integrating factor e^{kx}
• 3P = n/x: the integrating factor x^n
• 4A trig integrating factor and trig RHS
• 5Particular solution from an initial condition
• 6Show that a given function satisfies the ODE

Model answers (6)

• ✓Model answer 1
• ✓Model answer 2
• ✓Model answer 3
• ✓Model answer 4
• ✓Model answer 5
• ✓Model answer 6

Worked examples (6)

• 1Particular integral for a polynomial right-hand side
• 2Particular integral for an exponential right-hand side
• 3Particular integral for a trigonometric right-hand side
• 4Show that a given function is a particular integral
• 5The clash case — multiply the trial by x
• 6Full solution with two initial conditions

Model answers (6)

• ✓Model answer 1
• ✓Model answer 2
• ✓Model answer 3
• ✓Model answer 4
• ✓Model answer 5
• ✓Model answer 6

Worked examples (6)

• 1Forming the auxiliary equation
• 2Complementary function — distinct real roots
• 3Complementary function — repeated root
• 4Complementary function — complex conjugate roots
• 5Purely imaginary roots — the SHM case
• 6Non-monic equation with complex roots

Model answers (6)

• ✓Model answer 1
• ✓Model answer 2
• ✓Model answer 3
• ✓Model answer 4
• ✓Model answer 5
• ✓Model answer 6

Worked examples (6)

• 1Deriving the first-derivative transform
• 2Deriving the second-derivative transform
• 3Converting an Euler equation to constant coefficients
• 4Solving an Euler equation end-to-end
• 5Euler equation with a non-zero right-hand side
• 6Reduction of order with $u = y'$

Model answers (6)

• ✓Model answer 1
• ✓Model answer 2
• ✓Model answer 3
• ✓Model answer 4
• ✓Model answer 5
• ✓Model answer 6