1 Week 3 First-order ordinary differential equations (ODE) 1.Basic definitions 2.Separable ODEs 3.ODEs reducible to separable form 4.Linear first-order ODEs 5.Partial derivatives 6.Exact ODEs

2 1. Basic definitions ۞ An ordinary differential equation (ODE) is an equation involving an unspecified function y(x) and its derivatives. ۞ The order of an ODE is the order of the highest derivative involved. ۞ A linear ODE does not involve y 2, y 3, sin y, ( y' ) 2, ( y'' ) 3, etc.

3 1 st -order linear ODEs Example 1: 1 st -order nonlinear ODEs 2 nd -order linear ODEs 2 nd -order nonlinear ODEs

4 2. Separable ODEs Consider All such ODEs can be solved by integrating them w.r.t. x, Observe that x and y are now “separated” (hence, the name of this class of ODEs). Change the variable of integration on the l.-h.s.: x → y, and note that y' dx = dy,

5 Solution: Rewrite the ODE in the form Once the integrals in the previous equality are evaluated, we obtain a relationship between y and x – i.e. the solution! Now, ‘take apart’ dy and dx, move the latter to the r.-h.s. and integrate the equation Example 2:

6...and obtain (1) represents an implicit form of the general solution. In principle, you can omit one of the two constants of integration C 1,2 – but don’t omit both! If you do, you’ll end up with a particular solution instead of the general one (explanations to follow). (1) hence, hence... If the explicit solution is required, solve (1) for y by exponentiating,

7...hence, hence, where C = ±exp (C 1 – C 2 ) is an arbitrary constant (don’t forget the “ ± ”, otherwise C wouldn’t be fully arbitrary, it’ll be positive). (2) (2) Represents the general solution (its explicit form). It includes all possible particular solutions, corresponding to various values of C, e.g.

8 Initial-value problems In many applications, we are interested in a particular solution that satisfies some additional constraint, e.g. i.e. (3) is called an initial condition (even if x is not the time variable.) (3) Example 3: Solve the following initial-value problem:

9 Solution: Rewrite the ODE in the form hence, (this time, I’ve kept only one of the two constants of integration). Using the initial condition... (4)

10 Using the initial condition: hence, Substitute (4) into (5) to obtain an implicit solution of the initial- value problem, (5) or, explicitly, y x

11 ۞ Modelling is a mathematical description if natural (physical, biological, sociological, etc.) processes. Example 4: Radioactive decay y(t) = the amount of radioactive substance present at the time t. y' = the rate of radioactive decomposition. Experiments show that where k is a physical constant (can’t be calculated within the framework of our simple model, so should be measured). Note the minus on the r.-h.s. of (6) – it makes y(t) decrease with time ( y' < 0 ). (6)

12 Eq. (6) is separable, and its solution is Let’s impose an initial condition – say, (7) Then, (7) yields C = y 0 and (8) is the solution required, but let’s rewrite it in a more illustrative form, by introducing (8)

13 Now, (8) becomes hence, H is called the half-life. It’s the time in which half of the given amount of the radioactive substance decomposes. k

14 Certain ODEs are not separable, but can be transformed into separable form by a suitable substitution. 3. ODEs reducible to separable form Solve (i.e. find the general solution of) the following ODE: Example 5: Solution: Rearrange the ODE given in the form (9) (9) is not a separable ODE, but...

15 Let’s change the unknown y → u, where Observe that Now, (9) becomes hence, This ODE is separable and can be readily solved...

16 Note that the substitution u = y/x works for all ODEs of the form Example 6:...and can be readily solved: In terms of the original variable, y, the solution is (10) (11)

17 Solution: Let’s try to reduce our ODE to form (10) by changing the variables (x, y) → (X, Y) : so (11) becomes where a and b are undetermined constants (will be fixed later, as convenient). Observe that (12)

18 Choose a and b such that Now, (12) becomes which can be rearranged into This ODE is of form (10) – hence, it can be reduced to separable form by introducing u = Y/X.