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![Solution of an ODE
9
STM 604
Differential Equation
Any function 𝜙, defined on an interval I and possessing at least n derivatives
that are continuous on I, which when substituted into an nth-order ordinary
differential equation reduces the equation to an identity, is said to be a
solution of the equation on the interval.
From this definition, the interval I is variously called as interval of definition of
the solution and can be an open interval (𝑎, 𝑏), a closed interval [𝑎, 𝑏], an infinite
interval (𝑎, ∞), and so on.
Introduction to
Differential Equations
Definition
Classification
Solution of an ODE
Examples](https://image.slidesharecdn.com/unit1class-introductiontodifferentialequationsandfirst-orderdifferentialequations-240728020719-aa512af4/75/Introduction-to-Differential-Equations-and-First-Order-Differential-Equations-pptx-9-2048.jpg)
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Introduction to Differential Equations and First-Order Differential Equations.pptx
- 1. STM 604 –Differential Equation Introduction to Differential Equations 1 STM 604 Differential Equation ELLVAN M. CAMPOS, MST Faculty, Institute of Advanced Studies Introduction to Differential Equations
- 2. Lesson 1: Definitionand Terminologies 2 STM 604 Differential Equation Introduction to Differential Equations
- 3. Definition 3 STM 604 Differential Equation Definition Classification Solutionof an ODE Examples Consider this: 𝑦′′ + 2𝑦′ = 3𝑦 This is the same with 𝑓′′ 𝑥 + 2𝑓′ 𝑥 = 3𝑓 𝑥 or 𝑑2𝑦 𝑑𝑥2 + 2 𝑑𝑦 𝑑𝑥 = 3𝑦 Solution: function(s) Introduction to Differential Equations The difference of this kind of equation: 𝑥2 + 3𝑥 + 2 = 0 𝑥 = −2 or 𝑥 = −1 Solution: values(s) An equation containing the derivatives of one or more dependent variables with respect to one or more independent variables is a differential equation (DE).
- 4. Classification (by Type) If an equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable, it is said to be an ordinary differential equation (ODE). 𝑑𝑦 𝑑𝑥 + 5𝑦 = 𝑒𝑥 𝑑2𝑦 𝑑𝑥2 − 𝑑𝑦 𝑑𝑥 + 6𝑦 = 0 𝑑𝑥 𝑑𝑡 + 𝑑𝑦 𝑑𝑡 = 2𝑥 + 𝑦 4 STM 604 Differential Equation Introduction to Differential Equations Definition Classification Solution of an ODE Examples
- 5. Classification (by Type) 5 STM604 Differential Equation If an equation involving the partial derivatives of one or more dependent variables of two or more independent variables, it is said to be an partial differential equation (PDE). 𝜕2𝑢 𝜕𝑥2 + 𝜕2𝑢 𝜕𝑦2 = 0 𝜕2𝑢 𝜕𝑥2 = 𝜕2𝑢 𝜕𝑡2 − 2 𝜕𝑢 𝜕𝑡 𝜕𝑢 𝜕𝑦 = − 𝜕𝑢 𝜕𝑥 Introduction to Differential Equations Definition Classification Solution of an ODE Examples
- 6. Classification (by Order) Theorder of a DE is the order of the highest derivative in the equation. 𝑑2𝑦 𝑑𝑥2 + 5 𝑑𝑦 𝑑𝑥 3 − 4𝑦 = 𝑒𝑥 6 STM 604 Differential Equation Introduction to Differential Equations Definition Classification Solution of an ODE Examples
- 7. Classification (by Linearity) 7 STM604 Differential Equation An nth-order ordinary differential equation is said to be linear if F is linear in 𝑦, 𝑦′ , … , 𝑦𝑛−1 , in form: 𝑎𝑛 𝑥 𝑦 𝑛 + 𝑎𝑛−1 𝑥 𝑦 𝑛−1 + ⋯ + 𝑎1 𝑥 𝑦′ + 𝑎0 𝑥 𝑦 − 𝑔 𝑥 = 0 or 𝑎𝑛 𝑥 𝑑𝑛𝑦 𝑑𝑥𝑛 + 𝑎𝑛−1 𝑥 𝑑𝑛−1𝑦 𝑑𝑥𝑛−1 + ⋯ + 𝑎1 𝑥 𝑑𝑦 𝑑𝑥 + 𝑎0 𝑥 𝑦 = 𝑔 𝑥 Two Properties of Linear Differential Equation: (1) The dependent variable and all its derivatives are of the first degree; and (2) Each coefficient depends at most on the independent variable x. Introduction to Differential Equations Definition Classification Solution of an ODE Examples
- 8. Classification (by Linearity) 8 STM604 Differential Equation Samples 𝑦 − 𝑥 𝑑𝑥 + 4𝑥𝑑𝑦 = 0 𝑦′′ − 2𝑦′ + 𝑦 = 0 𝑑3𝑦 𝑑𝑥3 + 𝑥 𝑑𝑦 𝑑𝑥 − 5𝑦 = 𝑒𝑥 Non-Samples 1 − 𝑦 𝑦′ + 2𝑦 = 𝑒𝑥 𝑑2 𝑦 𝑑𝑥2 + sin 𝑦 = 0 𝑑4𝑦 𝑑𝑥4 + 𝑦2 = 0 Introduction to Differential Equations Definition Classification Solution of an ODE Examples
- 9. Solution of anODE 9 STM 604 Differential Equation Any function 𝜙, defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval. From this definition, the interval I is variously called as interval of definition of the solution and can be an open interval (𝑎, 𝑏), a closed interval [𝑎, 𝑏], an infinite interval (𝑎, ∞), and so on. Introduction to Differential Equations Definition Classification Solution of an ODE Examples
- 10. Example: Verification ofSolution 10 STM 604 Differential Equation Verify that the indicated function is a solution of the given differential equation on the interval (−∞, ∞). a) 𝑑𝑦 𝑑𝑥 = 𝑥𝑦 1 2; 𝑦 = 1 16 𝑥4 b) 𝑦′′ − 2𝑦′ + 𝑦 = 0; 𝑦 = 𝑥𝑒𝑥 Note: 𝑦 = 0 is a constant solution under the interval given. A solution of a differential equation that is identically zero on an interval I is said to be a trivial solution. Introduction to Differential Equations Definition Classification Solution of an ODE Examples
- 11. Example: Verification ofSolution 11 STM 604 Differential Equation Verify that the indicated function is a solution of the given differential equation on the interval (−∞, ∞). a) 𝑑𝑦 𝑑𝑥 = 𝑥𝑦 1 2; 𝑦 = 1 16 𝑥4 b) 𝑦′′ − 2𝑦′ + 𝑦 = 0; 𝑦 = 𝑥𝑒𝑥 Note: 𝑦 = 0 is a constant solution under the interval given. A solution of a differential equation that is identically zero on an interval I is said to be a trivial solution. Introduction to Differential Equations Definition Classification Solution of an ODE Examples
- 12. Lesson 2: Separable Variables 12 STM604 Differential Equation First-Order Differential Equations
- 13. Definition 13 STM 604 Differential Equation Definition Samples Afirst-order differential equation of the form 𝑑𝑦 𝑑𝑥 = 𝑔 𝑥 ℎ(𝑦) is said to be separable or to have separable variables. The equations 𝑑𝑦 𝑑𝑥 = 𝑦2𝑥𝑒3𝑥+4𝑦 and 𝑑𝑦 𝑑𝑥 = 𝑦 + sin 𝑥 are separable and nonseparable respectively. Method of Solution: Given that 𝑔 𝑥 𝑑𝑥 = ℎ 𝑦 𝑑𝑦, integrate the following separately. First-Order Differential Equations
- 14. Samples 1. Determine ifthe differential equation 1 + 𝑥 𝑑𝑦 − 𝑦𝑑𝑥 = 0 is separable. If it is, solve the DE using separation of variables. 14 STM 604 Differential Equation First-Order Differential Equations Definition Samples
- 15. Samples 2. Determine ifthe differential equation 𝑑𝑦 𝑑𝑥 = 3𝑥+𝑥𝑦2 𝑦+𝑥2𝑦 is separable. If it is, solve the DE using separation of variables. Answer: 𝑦2 + 3 = 𝑐(𝑥2 + 1) 15 STM 604 Differential Equation First-Order Differential Equations Definition Samples
- 16. Samples 3. Determine ifthe differential equation 𝑒2𝑦 − 𝑦 cos 𝑥 𝑑𝑦 𝑑𝑥 = 𝑒𝑦 sin 2𝑥 is separable. If it is, solve the DE using separation of variables, and find the specific equation if 𝑦 0 = 0. Answer: 𝑒𝑦 + 𝑦𝑒−𝑦 + 𝑒−𝑦 = 4 − 2 cos 𝑥 16 STM 604 Differential Equation First-Order Differential Equations Definition Samples
- 17. Samples 4. Determine ifthe differential equation 𝑑𝑦 𝑑𝑥 = 𝑦2 − 4 is separable. If it is, solve the DE using separation of variables. Answer: 𝑦−2 𝑦+2 = 𝑐𝑒4𝑥 17 STM 604 Differential Equation First-Order Differential Equations Definition Samples
- 18. Lesson 3: Linear Equations 18 STM604 Differential Equation First-Order Differential Equations
- 19. Definition 19 STM 604 Differential Equation Afirst-order differential equation of the form 𝑎1 𝑥 𝑑𝑦 𝑑𝑥 + 𝑎0 𝑥 𝑦 = 𝑔 𝑥 is said to be linear equation. When 𝑔 𝑥 = 0, the linear equation is said to be homogenous; otherwise, it is nonhomogenous. First-Order Differential Equations Definition Methods of Solution Samples
- 20. Solving a LinearFirst-Order Equation 20 STM 604 Differential Equation I. From 𝑎1 𝑥 𝑑𝑦 𝑑𝑥 + 𝑎0 𝑥 𝑦 = 𝑔 𝑥 , divide 𝑎1 𝑥 to obtain the standard form of the linear equation, given by 𝑑𝑦 𝑑𝑥 + 𝑃 𝑥 𝑦 = 𝑓(𝑥). II. From the standard form, identify 𝑃 𝑥 and then find the integrating factor 𝑒 𝑃 𝑥 𝑑𝑥 . III. Multiply the standard form of the equation by the integrating factor. The left-hand side of the resulting equation is automatically the derivative of the integrating factor and y: 𝑑 𝑑𝑥 𝑒 𝑃 𝑥 𝑑𝑥 𝑦 = 𝑒 𝑃 𝑥 𝑑𝑥 𝑓 𝑥 . IV. Integrate both sides of this last equation. First-Order Differential Equations Definition Methods of Solution Samples
- 21. Samples 1. Solve thedifferential equation 𝑑𝑦 𝑑𝑥 − 3𝑦 − 6 = 0. 21 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples
- 22. Samples 2. Solve thedifferential equation 𝑥 𝑑𝑦 𝑑𝑥 − 4𝑦 = 𝑥6𝑒𝑥. Answer: 𝑦 = 𝑥5 𝑒𝑥 − 𝑥4 𝑒𝑥 + 𝑐𝑥4 22 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples
- 23. Samples 3. Solve thedifferential equation 𝑥 𝑑𝑦 𝑑𝑥 − 𝑦 = 𝑥2 sin 𝑥. Answer: 𝑦 = 𝑐𝑥 − 𝑥 cos 𝑥 23 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples
- 24. Lesson 4: Exact Equations 24 STM604 Differential Equation First-Order Differential Equations
- 25. Differential of aFunction of Two Variables 25 STM 604 Differential Equation If 𝑧 = 𝑓(𝑥, 𝑦) is a function of two variables with continuous partial derivatives, it’s (total) differential is 𝑑𝑧 = 𝜕𝑓 𝜕𝑥 𝑑𝑥 + 𝜕𝑓 𝜕𝑦 𝑑𝑦. If 𝑧 = 𝑓 𝑥, 𝑦 = 𝑐, then 𝜕𝑓 𝜕𝑥 𝑑𝑥 + 𝜕𝑓 𝜕𝑦 𝑑𝑦 = 0. For example, given 𝑥2 − 5𝑥𝑦 + 𝑦3 = 𝑐, then 2𝑥 − 5𝑦 𝑑𝑥 + −5𝑥 + 3𝑦2 𝑑𝑦 = 0. So, is it equivalent to the differential 𝑑(𝑥2 − 5𝑥𝑦 + 𝑦3 ) = 0? First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 26. Definition 26 STM 604 Differential Equation Adifferential expression 𝑀 𝑥, 𝑦 𝑑𝑥 + 𝑁 𝑥, 𝑦 𝑑𝑦 is an exact differential if it corresponds to the differential of some function 𝑓 𝑥, 𝑦 . A first-order differential equation of the form 𝑀 𝑥, 𝑦 𝑑𝑥 + 𝑁 𝑥, 𝑦 𝑑𝑦 = 0 is said to exact equation if the expression on the left-hand side is an exact differential. First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 27. Theorem 27 STM 604 Differential Equation Let𝑀 𝑥, 𝑦 and 𝑁 𝑥, 𝑦 be continuous and have continuous first partial derivatives, then a necessary sufficient condition that 𝑀 𝑥, 𝑦 𝑑𝑥 + 𝑁 𝑥, 𝑦 𝑑𝑦 be exact differential is 𝜕𝑀 𝜕𝑦 = 𝜕𝑁 𝜕𝑥 . First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 28. Method of Solution 28 STM604 Differential Equation I. Given the differential equation 𝑀 𝑥, 𝑦 𝑑𝑥 + 𝑁 𝑥, 𝑦 𝑑 = 0, determine whether 𝜕𝑀 𝜕𝑦 = 𝜕𝑁 𝜕𝑥 holds. II. If 𝜕𝑀 𝜕𝑦 = 𝜕𝑁 𝜕𝑥 exists, express function 𝑓 for which 𝜕𝑓 𝜕𝑥 = 𝑀 𝑥, 𝑦 . Find 𝑓 by integrating 𝑀 𝑥, 𝑦 with respect to 𝑥, while holding 𝑦 constant: 𝒇 𝒙, 𝒚 = 𝑴 𝒙, 𝒚 𝒅𝒙 + 𝒈(𝒚), where the arbitrary function 𝑔(𝑦) is the “constant” of integration. III. Differentiate 𝑓 𝑥, 𝑦 = 𝑀 𝑥, 𝑦 𝑑𝑥 + 𝑔(𝑦) with respect to 𝑦 and assume 𝝏𝒇 𝝏𝒚 = 𝑵 𝒙, 𝒚 = 𝝏 𝝏𝒚 𝑴 𝒙, 𝒚 𝒅𝒙 + 𝒈′(𝒚). This gives 𝒈′ 𝒚 = 𝑵 𝒙, 𝒚 − 𝝏 𝝏𝒚 𝑴 𝒙, 𝒚 𝒅𝒙. IV. Integrate 𝒈′ 𝒚 with respect to 𝒚 and substitute the result to 𝑓 𝑥, 𝑦 = 𝑀 𝑥, 𝑦 𝑑𝑥 + 𝑔(𝑦). First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 29. Samples 1. Solve thedifferential equation 2𝑥𝑦𝑑𝑥 + 𝑥2 − 1 𝑑𝑦 = 0. 29 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 30. Samples 2. Solve thedifferential equation 𝑒2𝑦 − 𝑦 cos 𝑥𝑦 𝑑𝑥 + 2𝑥𝑒2𝑦 − 𝑥 cos 𝑥𝑦 + 2𝑦 𝑑𝑦 = 0. Answer: 𝑥𝑒2𝑦 − sin 𝑥𝑦 + 𝑦2 + 𝑐 = 0 30 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 31. Non-Exact Differential Equations 31 STM604 Differential Equation For non-exact DE, look for an integrating factor to multiply to the given DE. Given 𝑀 𝑥, 𝑦 𝑑𝑥 + 𝑁 𝑥, 𝑦 𝑑𝑦 = 0, • if 𝑀𝑦−𝑁𝑥 𝑁 is a function of 𝑥 alone, then an integrating factor for the given DE is 𝜇 𝑥 = 𝑒 𝑀𝑦−𝑁𝑥 𝑁 𝑑𝑥 • if 𝑁𝑥−𝑀𝑦 𝑀 is a function of 𝑦 alone, then an integrating factor for the given DE is 𝜇 𝑥 = 𝑒 𝑁𝑥−𝑀𝑦 𝑀 𝑑𝑦 First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 32. Samples 3. Verify isthe given DE 𝑥𝑦𝑑𝑥 + 2𝑥2 + 3𝑦2 − 20 𝑑𝑦 = 0 is not exact. Multiply the given DE by the indicated integrating factor 𝜇 𝑥, 𝑦 and verify that the new equation is exact. Solve the new DE. 32 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 33. Lesson 5: Solutionsby Substitution 33 STM 604 Differential Equation First-Order Differential Equations
- 34. Homogenous First-Order Differential Equations Afirst-order DE in the form 𝑀 𝑥, 𝑦 𝑑𝑥 + 𝑁 𝑥, 𝑦 𝑑𝑦 = 0 is said to be homogenous if both 𝑀 and 𝑁 are homogenous functions of the same degree. Thus, 𝑀 𝑡𝑥, 𝑡𝑦 = 𝑡𝛼𝑀(𝑥, 𝑦) and 𝑁 𝑡𝑥, 𝑡𝑦 = 𝑡𝛼𝑁(𝑥, 𝑦). Substitutions may be on the following ways: 1. Either of the substitutions 𝑦 = 𝑢𝑥 or 𝑥 = 𝑣𝑦, where 𝑢 and 𝑣 are new dependent variables. 2. For differential equation of the form 𝑑𝑦 𝑑𝑥 = 𝑓(𝐴𝑥 + 𝐵𝑦 + 𝐶) can be reduced to an equation by means of 𝑢 = 𝐴𝑥 + 𝐵𝑦 + 𝐶. These two ways reduces the differential equations to separation of variables. 34 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 35. Samples 1. Solve thedifferential equation 𝑥2 + 𝑦2 𝑑𝑥 + 𝑥2 − 𝑥𝑦 𝑑𝑦 = 0. 35 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 36. Samples 2. Solve thedifferential equation 𝑥 − 𝑦 𝑑𝑥 + 𝑥𝑑𝑦 = 0. Answer: 𝑥 ln 𝑥 + 𝑦 = 𝑐𝑥 32 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 37. Samples 3. Find theexplicit solution of the differential equation 𝑑𝑦 𝑑𝑥 = −2𝑥 + 𝑦 2 − 7; 𝑦 0 = 0. 32 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 38. Samples 4. Solve thedifferential equation 𝑑𝑦 𝑑𝑥 = 𝑥 + 𝑦 + 1 2. Answer: 𝑦 = tan 𝑥 + 𝑐 − 𝑥 − 1 32 STM 604 Differential Equation First-Order Differential Equations Definition Methods of Solution Samples Non-Exact DE
- 39. - End Thankyou! 33 STM 604 Differential Equation