Differential Equations

A differential equation is an equation involving the derivatives of an unknown function, and such equations describe how quantities change in physics, chemistry, biology and economics, from the cooling of tea to the growth of bacteria and money in a bank. In this chapter you will learn to find the order and degree of a differential equation, and to distinguish between its general solution, which contains arbitrary constants, and a particular solution, obtained from given initial conditions. You will verify that a given function is a solution of an equation. You will then learn three standard methods of solving first order, first degree equations. Equations with variables separable are solved by bringing all y terms to one side and integrating. Homogeneous equations are reduced to separable form by the substitution y = vx. Linear equations of the form dy/dx + Py = Q are solved using the integrating factor e^(integral of P dx). You will apply these methods to growth and decay, Newton's law of cooling and curves with given slopes.

Detailed Worksheet: Differential Equations Section A - Definitions (10 marks) 1. Define the order and the degree of a differential equation. When is the degree not defined? (2 marks) 2. Distinguish between the general solution and a particular solution of a differential equation. (2 marks) 3. What is a homogeneous function of degree n? Show that f(x, y) = x^2 + xy is homogeneous and find its degree. (2 marks) 4. Write the standard form of a first order linear differential equation and the formula for its integrating factor. (2 marks) 5. Verify that y = e^x + 1 is a solution of y'' - y' = 0. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the general solution of dy/dx = (x + 1)/(2 - y), where y is not equal to 2. (3 marks) 7. Find the particular solution of dy/dx = -4xy^2, given that y = 1 when x = 0. (3 marks) 8. Solve the linear differential equation dy/dx + 2y = sin x. (3 marks) 9. Solve the homogeneous equation dy/dx = (x + y)/x for x > 0, given that y = 2 when x = 1. (3 marks) 10. State the order and degree (if defined) of: (i) (y''')^2 + (y'')^3 + (y')^4 + y^5 = 0, (ii) y'' + (y')^2 + 2y = 0 and (iii) y' + sin(y') = 0. (3 marks) Section C - Diagrams (10 marks) 11. The family of curves y = Cx^2 is the general solution of dy/dx = 2y/x. Sketch the solution curves for C = -1, 1 and 2, and mark the particular curve passing through (1, 3). (4 marks) 12. Draw a flowchart to decide which method to use for a first order differential equation: variables separable, homogeneous (substitute y = vx) or linear (find integrating factor). (3 marks) 13. Sketch the graph of P = P0 e^(kt) for k > 0 (growth) and N = N0 e^(-kt) for k > 0 (decay), and mark the initial values. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Solve x dy/dx + 2y = x^2 for x not equal to 0, writing it in standard linear form and finding the integrating factor. (5 marks) 15. In a bank, principal increases continuously at the rate of 5% per year. Form the differential equation, solve it, and find in how many years Rs 1,000 will double (log 2 = 0.6931). (5 marks) 16. Show that the equation (x^2 + xy) dy = (x^2 + y^2) dx is homogeneous and solve it. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Identify the type of each equation before solving and show the constant of integration.