# Differential Equations Mathematics & Economics Math Problem (Math Problem Sample)

Differential Equations: Using Undetermined coefficients

source..1 u''+u'+2u= sin2t

The auxiliary equation is r2+r+2=0

Using the general formula

x=-b±b2-4ac2a

a = 1, b = 1, c = 2

x=-1±12-4(1)22(1)

x=-1±1-82

x=-1±1-82

x=-1±-72

x=-1+-72 or -1--72

∴If the roots are complex numbers, i.e r1= α+iβ , r2= α-iβ

Where α, β are real numbers, from the solution we have.

u(t)=eαt(c1cosβt + c2sinβt )

Hence -1±-72=α±iβ , where i= -1

-1±i72

-12±i72

Where α= -12 , β= 72

From the general solution,

2 ut=e-12tc1cos72t + c2sin7

3

4 2t u''+u=cosht where u0=1, u'0=4

The auxiliary equation is r2-1=0

r2=0+1

r2=1

Taking the square roots of both sides gives

r2=1

r= ±1

r1=1, r2=-1

From the general solution,

ut=c1er1t+ c2er2t

Substituting values of r1 =1 and r2= -1

ut=c1et+ c2e-t

Since gx= x2, we seek a particular solution (up)

Let the particular solution (up) = Ate2t+ Be-2t

Differentiating up,

up=Ate2t+ Be-2t

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