Learning Outcomes: (a) Distinguish between general & particular solutions. [State the degree & order.] (b) Solve separable differential equations [Use the boundary conditions to find a particular solution.] Solve first order linear differential equations by mean of an integrating factor. [Must show all the necessary steps.] (c) At the end of the lesson, students should be able to Differential Equation (D.E.) A differential equation (D.E.) is an equation with an unknown function and one or more of its derivatives. Order of D.E. The order of a D.E. is the order of the highest derivative in the equation. Degree of D.E. The degree of a D.E. is the power of the highest derivative in the equation. Differential Equation for function of single variable Highest Derivative term Order of D.E. order of the highest derivative Degree of D.E. power of the highest derivative 1st Order Degree 1 1st Order Degree 2 2nd Order Degree 1 2nd Order Degree 1 Remarks: Our syllabus only cover 1st order & degree 1 of D.E. 3.1 Separable Variables 3.2 First Order Linear Differential Equations C03 - SM025 Page 1
Solution of D.E. The solution of a D.E. is obtained by integrating the D.E. with respect to • [The solution of a D.E. is a function that satisfies the D.E.] the solution with the presence of an arbitrary constant. [ The solution is written as in term of (a) General Solution the general solution with its arbitrary constant is assigned to a particular value. (b) Particular Solution • 2 types of solutions of D.E. : [Obtained by solving the arbitrary constant in general solution] Remarks: Particular solution can be obtained without finding the general solution if the question does not request to find general solution. [especially the steps to obtain general solution is tedious.] • E.g. Differential Equation Solve by integrating with respect to General Solution Solve the arbitrary constant If given when Particular solution [ in general solution is assigned] Consider a D.E. in terms of and . The Solution of a Differential Equation C03 - SM025 Page 2
For a differential equation in term of , and • , it is called as the separable variable differential equation if the equation can be express in the form E.g. where is a function in term of and . (I) Consider the differential equation (II) Rearrange the equation to separate the variables (III) Integrate both sides of the equation in (II) (IV) General Solution: Write in term of and an unknown arbitrary constant. Particular Solution: Solve the unknown arbitrary constant and substitute into the general solution. Remarks: Since the question does not request for general solution, therefore the equation can be solved by finding its particular solution only . (without finding its general solution) Steps in solving Separable Variable Differential Equations : Separable Variable Differential Equations C03 - SM025 Page 3
(I) Consider the differential equation where is a function in term of and . (II) Rearrange the equation to separate the variables : (III) Integrate both sides of the equation in (II) : General solution: (IV) General Solution: Write in term of and an unknown arbitrary constant. Particular Solution: Solve the unknown arbitrary constant and substitute into the general solution. After general solution is obtained, For , Continue directly from step III (without finding general solution) For , Particular solution: [Given , i.e. ] Methods to Solve Separable Variable D. E. C03 - SM025 Page 4
Find the general solution for the differential equation Solution: where General Solution Example 1 [Example 1(a) in PowerPoint Notes Topic 3.1] C03 - SM025 Page 5
Find the particular solution for the differential equation given that when . Solution: For Example 2 [Example 2(a) in PowerPoint Notes Topic 3.1] C03 - SM025 Page 6
For a differential equation in term of , and • , it is called as the first order linear differential equation if the equation can be express in the form E.g. (I) Obtain the standard from: (II) Find the integrating factor where Steps in solving First Order Linear Differential Equations : (III) Multiply the equation in (I) by & obtain Express the LHS of the equation in (III) as the derivative of the product of the integrating factor and variable (IV) (V) Integrate both sides of the equation (IV) General Solution: Write in term of and an unknown arbitrary constant. Particular Solution: Solve the unknown arbitrary constant and substitute into the general solution. Remark: Do not add when finding First Order Linear Differential Equations C03 - SM025 Page 7
(I) Obtain the standard from: (II) Find the integrating factor where Remark: Do not add when finding & ignore modulus for most cases (III) Multiply the equation in (I) by & obtain (IV) Express the LHS of the equation in (III) as the derivative of the product of the integrating factor and variable , i.e. (V) Integrate both sides of the equation: General solution: (VI) General Solution: Write in term of and an unknown arbitrary constant. Particular Solution: Solve the unknown arbitrary constant and substitute into the general solution. After general solution is obtained, For , Continue directly from step V (without finding general solution) For , Particular solution: [Given , i.e. ] Method to Solve First Order Linear D. E C03 - SM025 Page 8
Find the general solution for the differential equation Solution: Multiply with integrating factor Integrating factor Example 3 C03 - SM025 Page 9
Find the particular solution for the differential equation given that . Solution: Multiply with integrating factor Integrating factor For Example 4 [Example 5 in PowerPoint Notes Topic 3.2] C03 - SM025 Page 10