Using method of variation find solution of y''+2iy'-y=e^x-2e^ix

Find the solution of x^2y''+xy'-3y=0 with y(0)=1,y'(0)=0 using laplace transform method?

Find the general integral of (x-y)p+ 9(y-x-z)q=z which passes through the circle x2+y2=1,z=1?

Give an example of a continuous function f with the property that∑[n=1,∞,f(n)] converges and ∫[0,∞,f(x),] diverges

Let f be real,continuously differentiable function on [a,b],f(a)=0=f(b) and ∫[a,b,f^n(x),]=1. Show ∫[a,b,xf(x)f'(x),]=-1/2

Show e^-x is uniformly continuous on R^1

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Let E be adense subset of ametric space X and let f be uniformly continuous function on E.Prove f has continuous extension from E to X

Is (a,b) open on R^2

Construct a component set of real numbers with a countable set of limit points

Construct a bounded set of a real numbers with exactly three limit points

An integral domain R with unity is a U.F.D if and only if every non zero ,non unit element is a Finite product of primes

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If W is the subspace of R^5 spanned by v1=(2,-2,3,4,-1) v2=-(1,1,2,5,2 ) v3=(0,0,-1,-2,3 ) v4=(1,-1,2,3,0) determineA(W?

Let T be a linear operator on finite dimensional vector space over F.Suppose that the minimal polynomial for T decomposes over F in to a product of linear polynomials.Show that there exits a diagonalizable operator D on V and a nilpotent operator N on V suchthat T=D+N and DN=ND

If f is a C'-mapping of an open set A is proper subset of R^n in to R^n and if f'(x) is invertible for every x belongs to A.Then prove that f(U) is open subset of R^n for every open set U is proper subset of A

Let{fn} be a sequence of continuous functions which converges uniformly to a function f on a set A.Then prove that lim[n:∞,]fn(xn)=f(x) for every sequence of a points xn belongs to A such that xn tends to x,and x belongs to A.Is the converse of this true?Explain with example