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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1506 Mathematics II Academic Year 2007-2008, Semester 2 Engineering Mathematics II: ............. JIN Chenyuan: Room S9A #02-03 Departments of Mathematics National University of Singapore 2 Science Drive 2, Singapore 117543 Republic of Singapore Office Number : 6516-8974 Email : [email protected] Homepage : http://jin.chenyuan.googlepages.com/ MA1506 Resources : http://jin.chenyuan.googlepages.com/ma1506
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Acknowledgements This is the slides and notes for my tutorial groups of MA1505. I’d be grateful for any comments or corrections. I would like to thank . . . My thanks also goes out to the Department of Mathematics January 2008
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Chapter 1 Tutorial 01 (Solution Notes) 1.1 Question 1 Solve the following differential equations: ( a ) x ( x + 1) y = 1 ( b ) (sec( x )) y = cos(5 x ) ( c ) y = e ( x 3 y ) ( d ) (1 + y ) y + (1 2 x ) y 2 = 0 Separable Equations If an O.D.E. is of the form N ( y ) dy dx = M ( x ) , then, we solve it with the following three steps: 1. First, we can separate dy and dx as follows: N ( y ) dy = M ( x ) dx. 2. The integrate on both sides, we have N ( y ) dy = M ( x ) dx + c, where c is a constant. 3. ( Sometimes this step is ignored. ) Then try to solve out the explicit formula of y from the above equation y = · · · · · · . 1
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2 MA1506 Tutorial 01 (Solution Notes) Remarks: 1. This is the method we have learnt in MA1505. 2. Although this solution only contains three steps (separation, integration, and explicitness), the difficulties may occur in each step. Actually, many solutions used in the lecture notes and tutorial problems are just trying to separate variables for special equations. So the solutions are not very hard. 3. If N ( y ) , the coefficient of y , is a nonzero constant, for example, N ( y ) = 1 , then life would be easier, we can omit the third step mentioned above. Check question (a) and (b). Graphmatica This is one of the efficient software to sketch mathematical graphs. You can download a free trial version from http://www.graphmatica.com . (one-month free trial) But to sketch the graph of a given differential equation, we need some initial conditions. Otherwise, the solution has a constant c , which is a family of curves instead of a single one. For example in we have a initial condition y (1) = 0 . 5, then we can find the value of c , and sketch the curve with the command “x(x + 1)dy = 1 { 1, 1/2 } ”. Remarks: 1. Sketch the curve of a given function BY HAND is required. That means in the exam, you cannot use the software or a calculator to help you. 2. The skill to get help from a software (MatLab, Maple, Mathematica, Graph- matica, etc.) is important in your Post-MA1506 career. You can learn the basic operations of MatLab during the lab section of MA1506.
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MA1506 Tutorial 01 (Solution Notes) 3 (a) x ( x + 1) y = 1 We can change the equation to y = 1 x ( x + 1) . Therefore, this is a separable O.D.E., so we can solve the equation by integrating on both sides.
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