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- 1. By: Prof. Yogiraj Mahajan HOD Science Dept. K. K. Wagh Polytechnic, Nashik-3
- 2. Differential Equations [ Marks-20] • Differential equation- Definition, order and degree of a differential equation. Formation of differential equation containing single constant. • Solution of differential equation of first order and first degree for following types Variable separable form . Equation reducible to variable separable form. Linear differential equation. Homogeneous differential equation. Exact differential equation. 2Prof. Yogiraj Mahajan
- 3. Differential Equations Definition : An equation consists of independent variable (x), dependent variable (y) and differential coefficient is called differential equation. ....,,, 3 3 2 2 dx yd dx yd dx dy 3Prof. Yogiraj Mahajan
- 4. Differential Equations 0y dx dy x 2 dx dy dx dy xy 0 2 2 2 xy dx dy dx yd xy dx dy yx 2 22 3 22 2 2 2 1 dx dy dx yd r 3 2 2 dx dy y dx yd 3 dx dy y dx dy dx dy x 5 3 2 2 dx dy y dx yd 4Prof. Yogiraj Mahajan
- 5. Differential Equations Order of the highest order derivative appearing in differential equation is called Order of differential equation. Degree of the highest order derivative when the derivatives are free from radical and fraction(-ve index) is called degree of a differential equation. Order of differential equation : Degree of a differential equation: 5Prof. Yogiraj Mahajan
- 6. Differential Equations 2 dx dy dx dy xy 0 2 2 2 xy dx dy dx yd xy dx dy yx 2 22 3 2 2 dx dy y dx yd Order – 1 , Degree – 1 Order – 2 , Degree – 2 Order – 2 , Degree – 1 Order – 1 , Degree – 2 6Prof. Yogiraj Mahajan
- 7. Differential Equations 0y dx dy x 3 22 2 2 2 1 dx dy dx yd r 3 2 2 dx dy y dx yd 3 dx dy y dx dy dx dy x 5 Order – 1 , Degree – 4 Order – 2 , Degree – 2 Order – 2 , Degree – 2 Order – 1 , Degree – 1 7Prof. Yogiraj Mahajan
- 8. Differential Equations Formation of Differential Equation: The process of eliminating arbitrary constant from the given relation by differentiation is called formation of differential equation. 8Prof. Yogiraj Mahajan
- 9. Form a differential equation whose solution is y = mx2 2 mxy x y dx dy 2 dx dy x m xm dx dy 2 1 2. 2 2 2 1 x y dx dy x x y m 9Prof. Yogiraj Mahajan
- 10. Differential Equations Relation between the dependent variable(y) and independent variable (x) [without their differential coefficients] along with arbitrary constant is called general solution of differential equation. The solution obtained from the general solution by giving particular values to arbitrary constants occurring in it is called particular solution of differential equation. 10Prof. Yogiraj Mahajan Solution of Differential Equation:
- 11. Since the process of solving of a differential equation recovers a function from knowing something about its derivative, it's not too surprising that we have to use integrals to solve differential equations. Differential Equations 11Prof. Yogiraj Mahajan
- 12. Differential Equations Solution of differential equation of first order and first degree for following types Variable separable form. Equation reducible to variable separable form. Linear differential equation. Homogeneous differential equation. Exact differential equation. 12Prof. Yogiraj Mahajan
- 13. Differential Equations An equation of the form Mdx + Ndy = 0 or where M and N are functions of x & y is called as differential equation of first order and first degree. 0 dx dy NM Differential Equation Of First Order And First Degree : 13Prof. Yogiraj Mahajan
- 14. Variable Separable Differential Equation [ By simple rearrangement separate x & y ] Mdx + Ndy = 0 Where M = f( x) fun. Of x only & N = g(y) fun. Of y only Solution is given by CNdyMdx Differential Equations 14Prof. Yogiraj Mahajan
- 15. 2 3xy dx dy dxx y dy 32 Cx y Cdxx y dy 2 2 3 2 1 3 Variable separable diff. eqn. CNdyMdx Differential Equations 15Prof. Yogiraj Mahajan
- 16. 011 dxydyx dyyxydxyx 111 2 0 2222 dxxyydyyxx 0e 2yx dyedx xy 0sec1tan3 2 ydyeydxe xx 2 3xy dx dy 0sin11 122 ydyxdxyx xdyyydxx 22 coscos 1yxxy dx dy 16Prof. Yogiraj Mahajan
- 17. 01 23 ydxxdyx 0 2222 dxxyydyyxx 0sec1tan3 2 ydyeydxe xx xyyx1 dx dy xy 0sincoscossin yx dx dy yx dx dy ya dx dy xy 2 3 2 1 x dxx y dy 0 11 22 dy y y dx x x 17Prof. Yogiraj Mahajan
- 18. Prof. Yogiraj Mahajan 18 Equation Reducible to Variable Separable Form If M or N contain ax + by + c then by substitution z = ax + by + c and Equation can be simplify to variable separable differential equation. Differential Equations dx dy ba dx dz
- 19. Prof. Yogiraj Mahajan 19 2 14 yx dx dy 22 a dx dy yx yx dx dy sin Solve: 342 12 yx yx dx dy dx dy yx 212cos 2
- 20. If y & are of first degree and not multiplied by each other then express equation as Where P & Q are functions of x only. Solution is given by Linear Differential Equation dx dy QPy dx dy CdxQeye PdxPdx Differential Equations Integrating factor IF = Pdx e 20Prof. Yogiraj Mahajan
- 21. 322 1 1 1 4 xx xy dx dy ecxxy dx dy coscot 1221 3 xxyx dx dy xx ti dt di 2sin106 1cossincos xxxy dx dy xx xy dx dy x tancos 2 0log2 2 xxy dx dy x xxy dx dy 2 costan 21Prof. Yogiraj Mahajan
- 22. If M & N are homogeneous expression in x and y and of same degree then express equation as Put Simplify so as to get variable separable diff. eqn. Solve it & put Homogeneous Differential Equation N M dx dy dx dv xv dx dy vxy & x y v Differential Equations 22Prof. Yogiraj Mahajan
- 23. y dx dy yx 2 2 yx dx dy xy yx yx dx dy 23 34 0 3 3 22 22 yx yx dx dy 0 332 dyyxydxx dxyxydxxdy 22 0 33 4 2 x yx dx dy y 0secsectan 22 dy x y xdx x y y x y x 23Prof. Yogiraj Mahajan
- 24. If M & N are functions of x & y and Solution is given by Exact Differential Equation x N y M CNdyMdx xfrom freeterm Minconst ykeep . Differential Equations 24Prof. Yogiraj Mahajan
- 25. 0cossecsincostancos 2 dyyxyxdxyxyx 0sectantan2 222 dyyyxxdxyyxy 0sin22 22 dyyxyxdxyxy x dx dy yxy 2 secsincos 0sectan3sec 2223 dyyxyxdxy 25Prof. Yogiraj Mahajan
- 26. 0 222222 ydybyxxdxayx 02362 2222 dyyxyxdxyxyx 0sinlogcos 1 1 dyyxxxdxy x y 0324 232 22 dyyxyedxxey xyxy 011 dy y x edxe y x y x 26Prof. Yogiraj Mahajan
- 27. Differential Equations •The most important application of integrals is to the solution of differential equations. •From a mathematical point of view, a differential equation is an equation that describes a relationship among a function, its independent variable, and the derivative(s) of the function. 27Prof. Yogiraj Mahajan
- 28. In Applications Differential equations arise when we can relate the rate of change of some quantity back to the quantity itself.
- 29. g dt xd dt dx dx d dt dv 2 2 The acceleration of gravity is constant (near the surface of the earth). So, for falling objects: the rate of change of velocity is constant Example (#1) Since velocity is the rate of change of position, we could write a second order equation: g dt dv 29Prof. Yogiraj Mahajan
- 30. 2 kvg dt dv Example (#2) Here's a better one -- with air resistance, the acceleration of a falling object is the acceleration of gravity minus the acceleration due to air resistance, which for some objects is proportional to the square of the velocity. For such an object we have the differential equation: rate of change of velocity is gravity minus something proportional to velocity squared or 2 2 2 dt dx kg dt xd 30Prof. Yogiraj Mahajan
- 31. In a different field: Radioactive substances decompose at a rate proportional to the amount present. Suppose y(t) is the amount present at time t. Example (#3) rate of change of amount is proportional to the amount (and decreasing) yk dt dy 31Prof. Yogiraj Mahajan
- 32. Other problems that yield the same equation: In the presence of abundant resources (food and space), the organisms in a population will reproduce as fast as they can --- this means that the rate of increase of the population will be proportional to the population itself: Pk dt dP 32Prof. Yogiraj Mahajan
- 33. ..and another The balance in an interest-paying bank account increases at a rate (called the interest rate) that is proportional to the current balance. So kB dt dB 33Prof. Yogiraj Mahajan
- 34. More realistic situations for the last couple of problems For populations: An ecosystem may have a maximum capacity to support a certain kind of organism (we're worried about this very thing for people on the planet!). In this case, the rate of change of population is proportional both to the number of organisms present and to the amount of excess capacity in the environment (overcrowding will cause the population growth to decrease). If the carrying capacity of the environment is the constant Pmax , then we get the equation: PPkP dt dP max 34Prof. Yogiraj Mahajan
- 35. and for the Interest Problem... For annuities: Some accounts pay interest but at the same time the owner intends to withdraw money at a constant rate (think of a retired person who has saved and is now living on the savings). 35Prof. Yogiraj Mahajan
- 36. Another application: According to Newton's law of cooling , the temperature of a hot or cold object will change at a rate proportional to the difference between the object's temperature and the ambient temperature. If the ambient temperature is kept constant at A, and the object's temperature is u(t), what is the differential equation for u(t) ? 36Prof. Yogiraj Mahajan
- 37. A differential equation of the form gives geometric information about the graph of y(x). It tells us: ),( yxf dx dy Geometry of Differential Equations If the graph of y(x) goes through the point (x,y), then the slope of the graph at that point is equal to f(x,y). ),( yxf dx dy 37Prof. Yogiraj Mahajan
- 38. Differential Equations Thanks for Silent leasing

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