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Application of Differential Equations .ppt
- 1. DIFFERENTIAL DIFFERENTIAL CALCULUS AND ITS CALCULUSAND ITS APPLICATIONS APPLICATIONS Shivani Sharma Shivani Sharma Assistant Professor Assistant Professor Department of Mathematics Department of Mathematics G.D.C. for Women, Kathua G.D.C. for Women, Kathua
- 2. Origin Origin Differential equationswas independently Differential equations was independently invented by English physicist Issac Newton invented by English physicist Issac Newton and German mathematician Gottfried and German mathematician Gottfried Leibnitz Leibnitz. .
- 3. DIFFERENTIAL DIFFERENTIAL CALCULUS IS CALCULUS IS EVERYWHERE EVERYWHERE Whereverwe study how something Wherever we study how something changes with respect to something else changes with respect to something else is an application of calculus is an application of calculus
- 4. DIFFERENTIAL EQUATION: DIFFERENTIAL EQUATION: An equation involving derivatives of one An equation involving derivatives of one or more dependent variables with respect or more dependent variables with respect to one or more independent variables is to one or more independent variables is called a differential equation. called a differential equation. In simple words, we can say that it is the In simple words, we can say that it is the rate of change of one or more quantities rate of change of one or more quantities with respect to another quantities. with respect to another quantities.
- 5. CLASSIFICATION CLASSIFICATION According tothe number of According to the number of independent variables involved. independent variables involved. According to their order of the According to their order of the highest derivative they are highest derivative they are classified as first order , second classified as first order , second order and so on. order and so on.
- 6. APPLICATIONS : APPLICATIONS : ITIS USED IN IT IS USED IN 1. 1. Determining the motion of a projectile, rocket, Determining the motion of a projectile, rocket, satellite or planet satellite or planet 2. 2. LCR circuits LCR circuits 3. 3. Studying the rate of decomposition of a Studying the rate of decomposition of a radioactive substance radioactive substance 4. 4. In mechanics In mechanics
- 7. IN MECHANICS IN MECHANICS We all read about NEWTONS SECOND LAW which states that the time We all read about NEWTONS SECOND LAW which states that the time rate of change of momentum of a body is proportional to the resultant force rate of change of momentum of a body is proportional to the resultant force acting on the body acting on the body In mathematical language, this law states that In mathematical language, this law states that d∕dt (mv)=KF d∕dt (mv)=KF It is a differential eqn in which the dependent variable is v and independent It is a differential eqn in which the dependent variable is v and independent variable is t. variable is t. Where m is the mass of the body ,v is its velocity ,F is the resultant force Where m is the mass of the body ,v is its velocity ,F is the resultant force acting upon it ,and K is constant of proportionality . acting upon it ,and K is constant of proportionality . If m is constant then If m is constant then mdv/dt=KF mdv/dt=KF a = KF/m a = KF/m F = Kma F = Kma And for K=1, And for K=1, F = ma F = ma
- 9. IN LCR CIRCUITS INLCR CIRCUITS A LCR circuit is a circuit consisting of an inductor ,capacitor, and a resistor along with A LCR circuit is a circuit consisting of an inductor ,capacitor, and a resistor along with electromotive force . electromotive force . The emf produces a flow of current in a closed circuit and this current produces voltage The emf produces a flow of current in a closed circuit and this current produces voltage drop across each resistor ,inductor,and capacitor. drop across each resistor ,inductor,and capacitor. We know that in a capacitor , We know that in a capacitor , The charge q(t)stored in a capacitor at any time t is directly The charge q(t)stored in a capacitor at any time t is directly proportional to the applied voltage proportional to the applied voltage q(t) = CV q(t) = CV V = q(t)/C V = q(t)/C Where C is the capacitance Where C is the capacitance Also the voltage drop across resistor Also the voltage drop across resistor By ohms law, By ohms law, V = IR V = IR also I = dq/dt also I = dq/dt So V =R dq/dt So V =R dq/dt That is the current produces is directly proportional to the applied voltage where R is the That is the current produces is directly proportional to the applied voltage where R is the resistance. resistance. The voltage drop across an inductor is given by The voltage drop across an inductor is given by V = LdI/dt V = LdI/dt V = Ld”q/dt V = Ld”q/dt where L Is the inductance. where L Is the inductance.
- 10. According to Kirchoff”ssecond law , According to Kirchoff”s second law , The sum of the voltages drop across The sum of the voltages drop across the resistor ,capacitor and inductor is the resistor ,capacitor and inductor is equal to the total emf in the closed equal to the total emf in the closed circuit . circuit . So, So, E(t)=L q”(t)+ Rq’(t)+ 1/Cq(t) E(t)=L q”(t)+ Rq’(t)+ 1/Cq(t) it is the second order differential eqn. it is the second order differential eqn.
- 11. Radioactivity Radioactivity decay decay The emissionof particles like alpha, beta particles or gamma The emission of particles like alpha, beta particles or gamma radiations by any radioactive element is called radioactive radiations by any radioactive element is called radioactive decay or radioactive disintegration. It is known that rate of decay or radioactive disintegration. It is known that rate of disintegration or decay of any radioactive material at any time disintegration or decay of any radioactive material at any time depends upon the number of atoms of the radioactive depends upon the number of atoms of the radioactive elements present at that time. elements present at that time. This can be expressed as a differential equation. This can be expressed as a differential equation. Rate of disintegration at time t Rate of disintegration at time t dN/dt = KN dN/dt = KN Where N is the no. of atoms present after time t and k is the Where N is the no. of atoms present after time t and k is the constant of proptionality. constant of proptionality. Since N is decresing dN/dt<0 Since N is decresing dN/dt<0
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- 13. Solution Solution Its solution canbe calculated by seperating the variables and Its solution can be calculated by seperating the variables and then on integrating we get, then on integrating we get, dN/dt = KN dN/dt = KN ∫ ∫dN/N = K∫dt dN/N = K∫dt Log N = Kt+I Log N = Kt+I Where I is the integration constant Where I is the integration constant If № is the number of particles present initially at time t=0 If № is the number of particles present initially at time t=0 Then, log№ = 0 +I Then, log№ = 0 +I So logN = Kt+log№ So logN = Kt+log№ logN-log№ = Kt logN-log№ = Kt logN/№ = Kt logN/№ = Kt From this eqn the value of the constant K can be obtained. From this eqn the value of the constant K can be obtained.
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