Differential equations

• The order of a differential equation is the highest derivative present in the equation.
• If a first order differential equation is of the form y' = f(x) g(y), then you can separate the variables and integrate, as in normal maths.
• If a first order differential equation is in the form y' + P(x)y = Q(x), use the integrating factor method.
• The integrating factor is e^ $\int$ P(x) dx.
• In the integrating factor method, multiply the equation by the integrating factor, write the LHS as y' (integrating factor y) and integrate both sides (just cancels the y' on the LHS).
• The auxiliary equation of a second order differential equation a y'' + b y + c y = k is a m^2 + b m + c = 0.
• Where the auxiliary equation has two real roots, a and b, the general solution will be y = A e ^ a x + B e ^ b x.
• Where the auxiliary equation has one repeated root, a, the general solution will be y = e ^ a x (A + B x).
• Where the auxiliary equation has two complex conjugate roots, p + i q and p - i q, the general solution will be y = e ^ p x (A sin (q x) + B cos (q x)).
• For homogeneous second order differential equations, find the auxiliary equation, and then the corresponding general solution.
• The particular integral is a function that satisfies the original differential equation, and is in the form of the function of the RHS of the equation.
• If the form of the particular integral is the same as that in the original equation, multiply the particular integral by x to prevent it all cancelling out.
• To find the particular integral, determine the form of it, differentiate it twice, substitute it into the original equation and compare coefficients.
• For non - homogenous second order differential equations, first find the corresponding homogenous general solution (known as the complimentary function) and then add the particular integral.
• When using boundary conditions, substitute the numbers given into the general solution, differentiate and equate the product to the value given.
• The motion of a particle moving with simple harmonic motion will satisfy the equation d^2x/dt^2 = - w ^2 x, where w is a constant. This is a second order homogenous differential equation.
• In SHM, acceleration can be represented by d^2x/dt^2 = v dv/dx.
• For a particle moving with damped harmonic motion, d^2x/dt^2 + k dx/dt + w ^2 x = 0 applies, which is a second order homogenous differential equation.
• When an auxiliary equation about SHM has two real roots, the particle experiences heavy damping and no oscillations are performed.
• When an auxiliary equation about SHM has one repeated root, the particle experiences critical damping and no oscillations are performed.
• When an auxiliary equation about SHM has complex roots, the particle experiences light damping and the amplitude of the oscillations will reduce exponentially over time.
• For a particle moving with forced harmonic motion, d^2x/dt^2 + k dx/dt + w ^2 x = f(t) applies. This is a second order non - homogenous differential equation.
• When coupling first order simultaneous differential equations, eliminate the function that is not in the end equation by making it the subject of the equation with the differential of the function wanted, differentiating, and substituting this into the other equation.
• When forming second order differential equations relating to damped and forced harmonic motion, draw a force diagram and use F = m a as F = m d^2x/dt^2.