What is Differential equations: Definition and 999 Discussions

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

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  1. Z

    Verify Green's Formula for a Simple DE

    These problems are from a practice problem set from MIT OCW's 18.03 "Differential Equations. Computing the convolutions ##t*1## and ##1*t## is straightforward. They both equal ##\frac{t^2}{2}##. Then, ##(q*1)(t)=\int_0^t q(\tau)d\tau## and ##(1*q)(t)=\int_0^t q(t-\tau)d\tau## which after a...
  2. C

    Solve (t^2-1)y'' +4ty'+2y=6t, given two particular solutions

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  3. C

    Solving differential equation (t^2-1)y''-6y=1

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  4. Z

    Given unit impulse response, obtain differential equation

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  5. Z

    Obtaining stability criterion for 2nd order ODEs in coefficient form

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  6. Z

    Conditions for the undetermined coefficients method to fail with sinusoidal input

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  7. Z

    Linearly independent functions with identically zero Wronskian

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  8. Z

    Why is this not a general solution to this nonlinear DE?

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  9. Z

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  10. chwala

    I Determining linear indepedence/dependence of a set of functions

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  11. Davidian

    Variation of parameter VS Undetermined Coefficients

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  12. berlinvic

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  13. Z

    Obtain a nonzero solution of ##y''-4y'+x^2(y'-4y)=0## by inspection

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  14. chwala

    I Determining equilibrium solutions to differential equations

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  15. Z

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  16. Z

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  17. M

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  18. P

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  19. T

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  20. L

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  21. T

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  22. S

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  23. S

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  24. matqkks

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  25. D

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  26. L

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  27. H

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  28. H

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  29. T

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  30. H

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  31. U

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  32. Haorong Wu

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  33. E

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  34. Vick

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  35. B

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  36. chwala

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  37. M

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  38. Maxicl14

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  39. V

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  40. The Bill

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  41. A

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    Hello. Considering this DE; $$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2 Now the solution for the initial value should be C = -44; And for x(t) I get ; $$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$ Now to get the biggest domain of definition I did this; $$ -44...
  42. V

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  43. J

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  44. W

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  45. chwala

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  46. U

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  47. B

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  48. galaxywanderer

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  49. A

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  50. M

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