Exercise practice: Differential Equations

5.1 How the exercises are arranged

The chapter has six exercise sets. Exercise 9.1 covers order and degree. Exercise 9.2 covers the formation of differential equations from families of curves. Exercise 9.3 covers the general and particular solutions of equations solved by separation. Exercise 9.4 covers the variable separable method, both the homogeneous and the non-homogeneous cases. Exercise 9.5 covers homogeneous equations in particular and Exercise 9.6 covers the first order linear equations. The miscellaneous exercise mixes the methods and adds applications.

5.2 Order, degree and formation

Exercise 9.1 asks for order and degree. Clear any radical or fraction in the derivatives first, then read off the highest derivative and its power. Exercise 9.2 gives a family with parameters and asks for the differential equation. Differentiate as many times as there are parameters, then eliminate the parameters between the family and the derivatives. For a family with one parameter, one derivative is enough; for two parameters, two derivatives are needed and the result is a second order equation.

5.3 The three solving methods

Exercise 9.3 items are solved by separation and then by using the initial condition. Exercise 9.4 items are separable: write the equation as a product and integrate both sides. Exercise 9.5 items are homogeneous: test the degrees, substitute y equal to v x, and solve the resulting separable equation. Exercise 9.6 items are linear: write the standard form, find the integrating factor and apply the formula. A worked linear case is dy by dx plus y over x equal x squared, whose factor is x and whose solution is y equal x cubed over four plus C over x.

5.4 Miscellaneous and applications

The miscellaneous exercise combines forming an equation with solving it, and adds word problems on growth, decay, cooling and geometry. For each word problem the sequence is fixed: name the variables, write the rate statement as a derivative equation, identify the method, solve, and use the given data to fix the constant. The final answer should be stated in the variables of the problem, not in the symbols of the calculation.

5.5 A worked separable exercise

A typical Exercise 9.4 question is dy by dx equal e to the x minus y. Rewrite as e to the y dy equal e to the x dx, integrate both sides to get e to the y equal e to the x plus C, and the general solution is y equal log of e to the x plus C. The initial condition, if given, fixes C. The whole question is two lines once the separation is written, which is why writing the separated form clearly matters.

5.6 A worked homogeneous exercise

A typical Exercise 9.5 question is dy by dx equal x plus y over x minus y. The right side is homogeneous of degree zero, so put y equal to v x. After substitution the equation becomes x dv by dx equal one plus v squared over one minus v, which separates as one minus v over one plus v squared dv equal dx over x. The integral of the left side is inverse tangent v minus half log of one plus v squared, and replacing v gives the implicit general solution.

5.7 A worked linear exercise

A typical Exercise 9.6 question is dy by dx plus y over x equal x squared. The integrating factor is x, the left side becomes the derivative of x y, and integration gives y equal x cubed over four plus C over x. The pattern recurs so often that the factor can be written down immediately when the coefficient of y is one over x.

5.8 How to use the exercise sets for revision

Do one question from each set in order and name the method before starting. If two consecutive questions from different sets use the same method, the second is a check on whether the recognition is automatic. The miscellaneous set is the real test, because the method is not named in the question. A final pass through the set with the method named aloud is the best revision for the exam, because it trains recognition rather than calculation.

5.9 Where the marks are in an exercise answer

The exercise answers in the textbook are written in the same order as the marking scheme: identify the method, write the standard form, carry out the substitution or the factor, integrate, and then apply the condition. A worked answer that skips the first two lines often looks shorter and cleaner, but it loses marks and it is harder to check. The four worked examples above all follow the marked order.