# maths_calculated_here.

Apr 1st, 2015
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Picture 1A TypedL( y'') - 9 (y') = L(-3)( 52 L {y}- 2s -1 ) - 9(5L {y}-2) = -3/5(52 -9s) L{y} -2s -1 +1= -3/5(52 -9s) L{y} = -3/5+ 2s -175(5-9) L {y}= (-3 +2s2-17)/5L{y} = Using partial fractions;2s2 - 17s -3= 2s2-17s-3= A s(5-9) + B(5-9) +cs2Let s=0-3= B(-9)Let s=96 =C (81)2s2 -17s- 3= As2 -9As +Bs-9B +Cs2Comparing coefficients of s22= A+C2= A + A=Picture 1B Typed ; y(0)=1 Take laplace transform on both sides:Note linearityZ[y"]- 9 Z[y'] +Z [3] = 0?????..eq.1Use laplace table to find:Z [y"] = S2 Y(s)- s.f(o)-f'(o)Z[y']= S.Y(s)-f(o)Z[1]= 1/sLet Y(s)=Z[Y]Plug these in to our eq.1 aboveS2Y(s)-s.f(o)-f'(o)-9s.Y(s)+9f(o)+ 3/5=0Plug in given initial conditions Y(o)=f(o)=1F'(o)=1=S2Y(S)-S-1-9S.Y(S)+9+3/5=0Solve for Y(S):(S2-9S) Y(S)= S+1-9-3/5=S-8-3/5Y(S)= Use partial fractions to break the above equationA(S-9) +BS(S-9) + CS2= S2-8S-3AS-9A+BS2-9BS+CS2=S2-8S-3B+C=1A-9B=-8-9A=-3???A=1/3, B=25/27 and C=2/27So Y(S)=Take the inverse laplace on both sides:Z-1[Y(S)]= Z-1 + Z-1 + Z-1Using table noteZ-1= 9tZ-1= SY(t)= e9tCheck: Y'(t)= 1/3+2/3e9t ????Y(o)=1Y"(t)= 6e9t????Y'(o)= 1/3+2/3=1Y"-9Y'+3=6e9t-3-6e9t+3=0Picture 1C TypedQ2. (Not clear)We first construct traditional matrixNow P4 since for 3 hours, there is 4x 45minutesFor grade to be most likely, look for row 3, which leads to that in 3hours Is about 0.451(Next statement is not clear at all)In class of 7, the stationary states wi

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