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![Linear differential equation
Definition
Any function on multiplying by which the differential
equation M(x,y)dx+N(x,y)dy=0 becomes a differential
coefficient of some function of x and y is called an
Integrating factor of the differential equation.
If μ [M(x,y)dx +N(x,y)dy]=0=d[f(x,y)] then μ is called
I.F](https://image.slidesharecdn.com/lineardifferentialequation-141128224852-conversion-gate01/75/Linear-differential-equation-2-2048.jpg)
Uploaded byPratik Sudra
Linear differential equation
This document defines and provides examples of linear differential equations. It discusses: 1) Linear differential equations can be written in the form P(x)y'=Q(x) or P(y)x'=Q(y), where multiplying both sides by an integrating factor μ results in a total derivative. 2) First order linear differential equations of the form P(x)y'=Q(x) have an integrating factor of e∫P(x)dx. The general solution is y(IF)=C. 3) Bernoulli's equation is a differential equation of the form P(x)y'+Q(x)y^n=R(x), where the general solution depends
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Linear differential equation
- 1.
- 2. Linear differential equation Definition Any function on multiplying by which the differential equation M(x,y)dx+N(x,y)dy=0 becomes a differential coefficient of some function of x and y is called an Integrating factor of the differential equation. If μ [M(x,y)dx +N(x,y)dy]=0=d[f(x,y)] then μ is called I.F
- 3. Case-1 +P(x)y=Q(x) Is called a first order linear differential equation. I.F=eP(x)dx The general solution is Y(I.F)=
- 4. Example-1 Theequation is Here P=1, Q=x I.F=e1dx =ex The solution is Y(I.F)= Yex= ex+c =xex-ex+c Y=x-1+ce-x
- 5. Case-2 +P(y)x=Q(y) Is Linear differential equation I.F= The general solution is X(I.F)=
- 6. Example-2 Theequation is Here P= , Q= I.F=e-logy= The solution is X(I.F)= X =
- 7. Bernoulli’s Equation Definition A differential equation is Case-1 The general solution is yn v(I.F)=
- 8. x2y6 y-5=x2 Putv=y-5 = -5y-6 + x2 =-5x2 Here P= , Q= I.F=e-5logx=x-5 The solution is v(I.F)= VX-5= dx +c VX-5= dx +c = x2 + c Y-5= x3 +cx5
- 9. Case-2 Thegeneral solution is v(I.F)=