differential equations

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differential equations

  1. 1. Y.SARATH BABUA.SHAMEER AHMEDPRESENTED BY:
  2. 2. ContentsDefinationTypesProperties and
  3. 3. DIFFERENTIAL EQUATIONS APPLICATIONS IN SCIENCE & ENGINEERINGDefinitionThese are the equations obtained eliminating of arbitraryconstants from f(x,y,z,a,b)=0 equation in which a,b areconstants.A differential equation is an equation involvingderivatives of an unknown function and possiblythe function itself as well as the independentvariable.Example4 2 2 3sin , 2 0, 0y x y y xy x y y x1st order equations 2nd order equation
  4. 4. Differential equations wasinvented by LEIBNITZIt was developed byJOHANN BERNOULLI
  5. 5.  The order of the differential equation is order ofthe highest derivative in the differential equation.Differential Equation ORDER32xdxdy09322ydxdydxyd36433ydxdydxyd123
  6. 6. Differential EquationDegree0322aydxdydxyd36433ydxdydxyd035322dxdydxyd113The degree of a differential equation is power ofthe highest order derivative term in the differentialequation.
  7. 7. Derivatives These Are TwoTypes1. An ordinary differential equations2. A partial differential equations0322aydxdydxyd32xdxdy02222yuxu04444tuxu1122
  8. 8. Newton’s law ofcoolingsTTdtdTEx: A murder victim is discovered and a lieutenant was to estimate the time ofdeath. The body is loacted in a room that body kept at a constant temperture of68◦F . The lieutenant arrived at 9.30P.M and measured the body temperture as94.4◦F at that time. Another measurement of the body temperture at 11P.M is89.2◦FAns : time of death 53.8 minutesRate Of Decay OfRadioactive Materialsy is the quantity present at anytime(t)dyydt─
  9. 9. Newtons Second Law In DynamicsLaw of naturalgrowth or decay N(t) is amount of substance at‘t’
  10. 10. In SchrodingerWave EquationThe Schrodinger equation is the name of the basic non-relativistic wave equation used in one version ofquantum mechanics to describe the behaviour of aparticle in a field of force. There is the time dependantequation used for describing progressivewaves, applicable to the motion of free particles. Andthe time independent form of this equation used fordescribing standing waves.
  11. 11. In Laplace transformsRL circuitL di/dt + Ri =E
  12. 12. Heat Equation InThermo DynamicsExample:
  13. 13. 101. Free falling stonegdtsd222. Spring vertical displacementkydtydm 22where y is displacement,m is mass andk is spring constanta=-g
  14. 14. JacobianProperties•If the Jacobian(J) value is zerothen the given two relations aredependent.•If the Jacobian(J) value is notzero then the given two relationsare independent.

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