Rescue Kit

Occasionally in this course, we’ll come across concepts that you’ll have studied before, perhaps a couple of years ago, that you may not have used too often in the intervening time. Everyone has the odd topic that they’ve gone a bit rusty on, so on this page, I’m going to put links that might help you revise any such concepts that crop up in the lectures.

First order linear ODEs: integrating factor

A common theme in many of the methods we’ll use to solve partial differential equations (PDEs) is to reduce them to ordinary differential equations (ODEs), which then need to be solved.

An ODE of the form

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x) \]

may be solved by multiplying throughout by an integrating factor of \(e^{\int P(x)\mathrm{d}x}\), yielding

\[ e^{\int P(x)\mathrm{d}x}\frac{\mathrm{d}y}{\mathrm{d}x}+e^{\int P(x)\mathrm{d}x} P(x)y=e^{\int P(x)\mathrm{d}x} Q(x), \]

or equivalently

\[ \frac{\mathrm{d}}{\mathrm{d}x}\left\{e^{\int P(x)\mathrm{d}x}y\right\}=e^{\int P(x)\mathrm{d}x}Q(x). \]

Note that this equivalence can be seen by applying the product rule to the expression in curly brackets. Doing so will result in the previous equation.

Integrating both sides with respect to \(x\) and rearranging yields the solution \(y\).

Solving first order linear ODEs - integrating factor

Useful resources

First order ODEs: separation of variables

An ODE of the form

\[ \frac{\mathrm{d}y}{\mathrm{d}x}=f(x)g(y) \]

may be solved by “rearranging” to get

\[ \int\frac{\mathrm{d}y}{g(y)}=\int f(x)\mathrm{d}x. \]

Note that this is a slight abuse of notation as it treats \(\frac{\mathrm{d}y}{\mathrm{d}x}\) as though it were a fraction, which strictly speaking it is not (recall first principles definitions, for example the limit definition of the derivative). Nevertheless, this can be rigorously justified.

Both sides can now (hopefully) be integrated and the resulting expression rearranged to obtain a solution.

Useful resources

Second order linear ODEs

ODEs of the form

\[ a\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+b\frac{\mathrm{d}y}{\mathrm{d}x}+cy=f(x), \]

where \(a, b, c\in\mathbb{R}\) and \(f\) is a known function, have solutions that are the sum of two parts:

\[ y(x)=y_c(x)+y_p(x). \]

The complementary function, \(y_c\), is the general solution to the homogeneous problem

\[ a\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+b\frac{\mathrm{d}y}{\mathrm{d}x}+cy=0, \]

while the particular integral, \(y_p\), is any solution of the original problem

\[ a\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+b\frac{\mathrm{d}y}{\mathrm{d}x}+cy=f(x). \]

Constructing the complementary function

To construct the complementary function, we consider the quadratic equation

\[ am^2+bm+c=0. \]

This is called the auxiliary equation. Then:

  • if the auxiliary equation’s roots are real and distinct, \(m_1\) and \(m_2\), say, then the complementary function is of the form

\[ y_c(x) = Ae^{m_1 x}+Be^{m_2 x}. \]

  • if the auxiliary equation has a single repeated root, \(m\), say, then the complementary function is of the form

\[ y_c(x) = (A + Bx)e^{m x}. \]

  • if the auxiliary equation has complex conjugate roots, \(m_\pm=\alpha \pm i\beta\), say (for some \(\alpha,\beta\in\mathbb{R}\)), then the complementary function is of the form

\[ y_c(x) = e^{\alpha x}(A\cos(\beta x)+B\sin(\beta x)). \]

In all cases above, \(A\) and \(B\) are arbitrary constants.

Constructing the particular integral

This is the trickier bit. We won’t need to do this very often in this module, but if you’d like a reminder of how, this video might be useful:

Second order linear ODEs with constant coefficients

Useful resources

Parameterisation of curves

In the general case of the method of characteristics, we will parameterise curves. This essentially means that we write the \((x,y)\) co-ordinates of the points on the curve as \((x(t),y(t))\), where \(x(t)\) and \(y(t)\) are functions of a parameter \(t\). As \(t\) varies, the point moves along the curve.

If you want to read further about curve parameterisation, the external resource below is useful.

Useful resources