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Determine the Green, s function and express the solution as a definite integral

-y''=f(x), y(0)=0, y'(1)=0


Determine the Green function and express the solution as a definite integral

-(y''+y)=f(x), y'(0)=0, y(1)=0


Find the eigen value and eigen function of homogeneous integral equation

y(x)= 𝜆"\\int"K(x,t)y(t)dt where K(x,t)= sinx sin(t-1) -𝜋<=x<=t

sint sin(x-1) t<=x<= 𝜋



A tank having a capacity of 1000 liters, initially contains 400 liters of sugar water having a concen-

tration of 0.2 Kg of sugar for each liter of water. At time zero, sugar water with a concentration of

50 gm of sugar per liter begins pumped into the tank at a rate of 2 liter per minute. Simultaneously,

a drain is opened at the bottom of the tank so that the volume of the sugar-water solution in the

tank reduces 1 liter per minute. Determine the following:


integral curve of dx/y^2+yz+x^2= dy/y^2-xz+x^2=dz/z(x+y)
state and prove the necessary conditions for the existence of maximum value and minimum value?
state and prove the necessary conditions for the existence of maximum value and minimum value?

Find the general solution of the differential equation y" + 16y = 2 sec(4t) tan(4t)

Select one:

O none of the given answers is true

y = ci cos(4t) + c2 sin(4t) + ,t cos(4t) - sin(4t) In | cos(4t) |


y = ci cos(4t) + c2 sin(4t) - t cos(4t) + sin(4t) In | cos(4t) |

y = ci cos(4t) + c2 sin (4t) -

7t sin(4t) - - cos(4t) In | sin(4t) |

y = ci cos(4t) + C2 sin(4t) + ,t sin(4t) + - cos(4t) In | sin(4t)|



If yp = uj (t) cos(2t) + u2(t) sin(2t) is a particular solution of the differential equation

y" + 4y= csc(2t) cot(2t), then u, (t) =

Select one:

O none of the given answers is true

O Lt + - cot(2t)


O _ In | sin(2t) |

O

_ In | sin(2t) |

O

cot (2t)



Reduce into canonical form
4Uxx+5Uxy+Uyy+Ux+Uy=0