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

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