- •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
39. y = (1+ x2 )(arctg2 x +C)2 . |
40. 3x 2 y y3 C . |
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45. x3 (1 ln y) y 2 C . 46.100e-0,0344t; 310 years. 47. ln C
48. 2xy = 9. 49. x = – |
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41.x 2 y 3y 2 x y3 C .
44.x3 y -cos x -sin y = C .
x2 y2 arctg y 0 .x
T1. Individual tasks
1.1. Solve the separable differential equations.
1.1.1. 3(x2 y 2 x2 )dx (2y x3 y)dy 0 . |
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xey y ¢ = e2 y +1. |
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1.1.3. y(4 x2 )dy |
1 y 2 dx 0 . |
1.1.4. |
y x4 x y . |
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1.1.5. (x 2 y 2 x 2 )dx ( y 2 |
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1.1.6. |
1 x2 dy ydx 0 . |
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1.1.7. x(1 y |
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1.1.8. |
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1.1.9. cos yex dx +(1+e2x ) sin ydy = 0 .
1.1.11. y x cos2 y sin 2 x 0 .
1.1.13. |
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y(1 2x) y (1 2x) . |
1.1.17. |
y(1 x2 ) y x(1 y 2 ) 0 . |
1.1.19. |
ydy -x sin x 9- y2 dx = 0 . |
1.1.21.1+x2 dy = tg ydx .
1.1.23. (1+x2 )dy = 1- y-2 dx . 1.1.25. tg xdy + 1- ydx = 0 .
1.1.27. xx¢ = y cos( y2 ) 1+x2 .
1.1.29. e-sin y dx = x cos y ln xdy .
1.1.10. xy ¢-4 = y2 . 1.1.12. yy (1 x 2 ) 1 y 2 . 1.1.14. (x 1) yy 1 y 2 . 1.1.16. x2 yy ¢+3 = y2 .
1.1.18. y ¢ = y3 -1 . y2 x2
1.1.20.(1 x3 ) y x2 (1 y) .
1.1.22.x xy 2 y (4 x2 ) .
1.1.24.(1 x) yy e y2 .
1.1.26.y ¢ = 2 y ctg x .
1.1.28.e1/ y y y 2 x2 ln x .
1.1.30.y 2 dx x ln xdy 0 .
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1.2. Solve the homogeneous differential equations.
1.2.1. 2 y ¢ = e |
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1.2.2. |
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1.2.3.y x 2y 1.
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1.2.5. (x y)dx ( y 2x)dy 0 . 1.2.7. ( y 2x)dx ( y 2x)dy 0 . 1.2.9. (3y 2 2x2 )dx ( y 2 x2 )dy .
1.2.11. y xx 42yy .
1.2.13. (2x 3y)dx ( y 2x)dy .
1.2.15. xy x sin xy y .
1.2.17. 4xy y 2 x(x y) y . 1.2.19. (x2 y 2 ) y 2( y 2 xy) . 1.2.21. 2x2 y 2 2x 2 y .
1.2.23. xy y(ln2 ( y / x) 1) .
1.2.25. (x - y) y¢ = x +2y .
1.2.27. xy sin( y / x) x y sin( y / x) .
1.2.4. |
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(x2 y 2 )dx x 2 dy 0 . |
1.2.8.2xydy (x2 y2 )dx .
1.2.10.(2 y -x)dx = (3x + y)dy .
1.2.12.xy¢- y = x ctg xy .
1.2.14. 2 yxy ¢ = x2 + y2 .
1.2.16.xy xe2 y / x y .
1.2.18.(x 2y)dy (2x y)dx 0 .
1.2.20.(2x2 y 2 ) y y 2 2xy .
1.2.22. xy y x sec(y / x) . 1.2.24. x2 y ¢ = y2 -2x2 .
1.2.26.xy y xe y / x .
1.2.28.x2 y y 2 xyy .
1.2.29. xy y |
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1.2.30. xy y 2 (3x2 |
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1.3. Solve the linear differential equations. |
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1.3.1. (1 x2 ) y 2xy x. |
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1.3.2. (4 x2 ) y 2xy (4 x2 )2 . |
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1.3.3. y y / x x cos x . |
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1.3.5. (9 x |
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1.3.7. y ¢- |
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1.3.8. y sin x y cos x cos2 x . |
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1.3.9. x2 y y e1/ x x . |
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1.3.10. (x3 1) y 3x2 y x 1. |
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1.3.17. xy y x ln x 2 . |
1.3.18. xy 2y x |
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1.3.19. y sin x 3y cos x sin 2x . |
1.3.20. y 2 y tg x tg x sin x . |
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1.3.21. (4 x2 ) y 2xy 1 . |
1.3.22. xy 2y x2 |
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1.3.23. x2 y e1/ x sin(1/ x) y . |
1.3.24. xy y sin x . |
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1.3.25. y ¢- y tg x = cos-3 x . |
1.3.26. y y sin x 0.5sin 2x . |
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1.3.27. y ¢- y tg x = cos2 x . |
1.3.28. y ¢+ y tg x =1/ cos x . |
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1.3.29. y ¢+ y ctg x =sin2 x . |
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1.4. Solve the Bernoulli’s differential equations. |
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1.4.1. xy 2y x2 |
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1.4.2. y ¢- y ctg x = y2 . |
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1.4.3. |
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1.4.5. (1 x2 ) y xy y2 .
1.4.7. y y xy3 . x 1
1.4.9. xy y y4 . 1.4.11. xy y xy3 . 1.4.13. xy y xy2 sin x .
1.4.15. y xy y 2 ln 2 x .
1.4.17. (4 x 2 ) y 2xy x / y . 1.4.19. 3xy y (x2 1) y 2 .
1.4.21. 2xy y x2 y 1 .
1.4.4.xy y x2 y4 .
1.4.6.(1 x2 ) y xy x2 y 2 .
1.4.8.xy 2 y y 2 ln x .
1.4.10.4xy 3y ex x4 y5 .
1.4.12.y (3y 2xy) x .
1.4.14.( yx2 2) ydx xdy 0 .
1.4.16.y ¢+ y ctg x = y3 cos x .
1.4.18.3y 2 y y3 / x x 1 .
1.4.20.2(1 x2 ) y 2xy x / y .
1.4.22. |
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