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differential equations

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  • 1. Y.SARATH BABUA.SHAMEER AHMEDPRESENTED BY:
  • 2. ContentsDefinationTypesProperties and
  • 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. Differential equations wasinvented by LEIBNITZIt was developed byJOHANN BERNOULLI
  • 5.  The order of the differential equation is order ofthe highest derivative in the differential equation.Differential Equation ORDER32xdxdy09322ydxdydxyd36433ydxdydxyd123
  • 6. Differential EquationDegree0322aydxdydxyd36433ydxdydxyd035322dxdydxyd113The degree of a differential equation is power ofthe highest order derivative term in the differentialequation.
  • 7. Derivatives These Are TwoTypes1. An ordinary differential equations2. A partial differential equations0322aydxdydxyd32xdxdy02222yuxu04444tuxu1122
  • 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. Newtons Second Law In DynamicsLaw of naturalgrowth or decay N(t) is amount of substance at‘t’
  • 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. In Laplace transformsRL circuitL di/dt + Ri =E
  • 12. Heat Equation InThermo DynamicsExample:
  • 13. 101. Free falling stonegdtsd222. Spring vertical displacementkydtydm 22where y is displacement,m is mass andk is spring constanta=-g
  • 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.