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Unit IV- Differential Equations- IA. Variable separable formB. Exact differential equationsC. Homogeneous differential equationsD. Linear differential equationsE. Bernoulli’s Differential equationsF. Applications of differential equationsA. Variable separable form:Exercise 4.1Solve the following DE by separating their variables:1. xdy=ydx2. dydx=1+y21+x23. 1-x2dy+1-y2dx=04. dydx=e3x-y5. dy/dx = xy+x+y+16. dy/dx = cos(x+y) put x+y=v7. dy/dx = (x+y)/(x-y)put y=vx8. (x-y)2 dy/dx =1put v=x-y9. dy/dx = ( y + sqrt(x2-y2))/xput y=vx B. Exact Differential Equations: A DE of the form Mdx+Ndy=0 is said to be exact if ?M?y=?N?x.The general solution of an exact de is given by keeping y constantMdx+terms without xNdy=cExercise 4.2Solve the following differential equations:1. (2x+3cosy)dx+(2y-3xsiny)dy=02. (2x2+6xy-y2)dx+(3x2-2xy+y2)dy=03. (3x2+6xy2)dx+(6x2y+4y2)dy=04. (x+ycosx)dx+sinxdy=05. (2x+3y+4)dx+(3x+4y+5)dy=0C. Homogeneous differential equations:If any homogeneous DE is non-exact then it can be made exact by multiplying 1Mx+Ny [called as ‘integrating factor’].Exercise 4.3Solve the following homogeneous DE.1. (3xy2-y3)dx+(xy2-2x2y)dy=0.2. (x4+y4)dx-xy3dy=0.3. (x2y-2xy2)dx-(x3-3x2y)dy=0.4. (x2+y2)dx+8xydy=0.5. (xy+y2)dx-(x2-xy)dy=0.D. Linear differential equations (LDE): A DE of the form dydx+Py=Q , where P and Q are functions of x only (or may be constants), is known as linear DE. Its general solution is given by yePdx=Q.ePdxdx+cExercise 4.41. dydx +ycotx=cosecx.2. (x2+1)dydx+2xy=2x.3. xlogxdydx+y=2logx.4. cosxdydx+ysinx=1.5. xdydx-y=x2.E. Bernoulli’s Differential Equations (BDE): A DE of the form dydx+P.y=Q.yn , where P and Q are constants or functions of x only, is called Bernoulli’s DE. Steps to solve BDE 1. Divide both side by yn. 2. Put 1yn-1=v and 1yndydx=11-ndvdx and simplify. 3. Use the formula of LDE.Exercise 4.51. dydx+yx=y3. [ans. 1x2y2=2x+c]2. cosxdydx+ysinx=y4sin2x. [ans. cos3xy3=-32cos4x+c]3. 3dydx-3y=-αy2. [ans. 1y=α3+ce-x]F. Applications [ in simple electrical circuits] E= Ldidt + RiQ.1 A resistance of 100Ω and an inductance of 0.1H are connected in series with battery of 20V. Find current in circuit at any time ‘t’.Q.2 In a network circuit of R-L series R=50Ω and L=10H, a constant voltage 100V is applied at t=0 by closing the switch. Find the current in the circuit at t=0.5sec. ................
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