- •Module 3
- •Topic 1 .Differential equations of the first order and the first degree
- •Typical problems
- •Self-test and class assignments
- •Individual tasks
- •1.1. Solve the separable differential equations.
- •1.2. Solve the homogeneous differential equations.
- •1.3. Solve the linear differential equations.
- •1.4. Solve the Bernoulli’s differential equations.
- •1.5. Find the general solution and also the particular solution through the point written opposite the equation.
- •1.6. Solve the exact differential equations.
- •Various types of differential equations with appropriate substitution will be considered in the following articles (see table 3.1).
- •Table 3.1
- •Consider other types of differential equations with appropriate substitution for reduction of order:
- •1) a differential equation
- •Typical problems
- •Self-tests and class assignments
- •Answers
- •Table3.2
- •Table 3.4
- •Examples of typical problems
- •Class and self assignments
- •Answers.
- •3.2. Find the general solutions of linear homogeneous equations.
- •3.3. Find general the solutions of linear homogeneous equations with right part of special form.
- •3.4. Solve Cauchy’s test for equations of the second order.
- •3.5. Solve the equations using the Lagrange’s method.
- •Examples of typical problems solving
- •Tests for general and self-studying
- •Answers
3.4.16. |
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3.4.17. |
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3.4.18. |
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3.4.19. |
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3.4.20. |
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3.4.21. |
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3.4.22. |
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3.4.23. |
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3.4.24. |
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3.4.25. |
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3.4.26. |
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3.4.27. |
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3.4.28. |
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3.4.29. |
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3.4.30. |
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3.5. Solve the equations using the Lagrange’s method. |
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3.5.1. |
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3.5.2. y |
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3.5.3. |
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3.5.4. y 4 y tg2 2x . |
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3.5.5. |
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3.5.6. y |
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3.5.7. y 4y 4 y e2x ln(x2 |
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3.5.11. |
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3.5.13. |
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3.5.8.y 2 y y e x ln(x2 4) .
3.5.10.y y sin13 x .
3.5.12. y |
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3.5.14. y 2y |
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3.5.15. |
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3.5.17. |
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3.5.19. y |
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3.5.23. y 3y 2y cos(e x ) . |
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3.5.27. y 2y y e x ln x . |
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3.5.16.y 3y 2y sin(e x ) .
3.5.18.y 4y 4y x x1 e2x .
3.5.20.y 2y y e x arctg x .
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3.5. 24. |
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3.5.26. |
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3.5.28. |
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Topic 4. Simultaneous differential equations
Normal simultaneous differential equations. Methods of eliminating and integrated combinations of solving simultaneous differential equations in normal form. Simultaneous differential equations with constant coefficients. Generalized Euler’s method.
Bibliography: [2, 3, chapter 3.3], [3, chapter 8, §6], [4, chapter 8, §26], [6, chapter 11, item 11.5], [7, chapter 13, §§29–30], [8, 2 chapter, §§6].
T 4. General theoretical information
4.1. Normal simultaneous differential equations
Simultaneous differential equations of 1st kind (or first order):
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where y1 (t), y2 (t), , yn (t) are the unknown functions, t is independent
variable containing system in a normal form or system, which can be obtained concerning derivatives of unknown functions yi (t) , i 1, 2, , n .
The solution of the system (3.33) on interval ( a, b ) is the set of n of continuously differentiated functions
y1 1 (t), y2 2 (t), ..., yn n (t), which transform each equation of this system into equality.
Cauchy’s test for system (3.33) is to find such a solution, which depends on the initial condition:
y1 (t0 ) a1 , y2 (t0 ) a2 , ..., yn (t0 ) an , where a1 , a2 , ..., an – are the real values.
For solving simultaneous differential equations in normal form we use the following methods:
1)method of eliminating;
2)method of integrated combinations.
General form of the method of eliminating. After simultaneous equations differentiating and eliminating all unknown functions yi (x) , except one, we
obtain a differential equation of n order concerning the one function (for example, y1 ). After integration of this equation it is possible to find other
unknown functions.
The meaning of the method of integrated combinations is to make socalled integrated combinations from the equation of the given system with the help of arithmetical operations, that is that the equation concerning some new
function u u(t, y1 , y2 , .., yn ), which can be easily integrated.
4.2. Euler’s method for solving simultaneous differential equations with constant coefficients
The system of normal linear differential equations with constant coefficients we can name such system
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where |
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called homogeneous, in other case – right hand member not zero.
Let’s consider algebraic method of solving linear homogeneous simultaneous differential equation (generalized Euler’s method).
For example;
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General solution of the system (3.35) can be written as
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(3.36) in system (3.35) we obtain homogeneous system of linear algebraic
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The obtained system should be not equal to zero. So,
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