save_figs = F

# Question 9:
#
A = matrix( c( 1, 1, -2, 1 ), nrow=2, ncol=2, byrow=T )
b = c( 1, 1 )
x = solve( A, b )
print(x)
ts = seq( 0, 10, length.out=100 )
#ys = x[1] * exp(r1 * ts) + x[2] * exp(r2 * ts)
#plot(ts, ys)


# Question 10:
#
A = matrix( c( 1, 1, -3, -1 ), nrow=2, ncol=2, byrow=T )
b = c( 2, -1 )
print( solve( A, b ) )


# Question 11:
#
A = matrix( c( 1, 1, 1/2, 1/3 ), nrow=2, ncol=2, byrow=T )
b = c( 4, 0 )
print( solve( A, b ) )


# Question 13:
#
r1 = ( -5 - sqrt(13) ) / 2
r2 = ( -5 + sqrt(13) ) / 2
A = matrix( c( 1, 1, r1, r2 ), nrow=2, ncol=2, byrow=T )
b = c( 1, 0 )
x = solve( A, b )
print( x )
ts = seq( 0, 5, length.out=100 )
ys = x[1] * exp(r1 * ts) + x[2] * exp(r2 * ts)
if( save_figs ){ postscript("../../WriteUp/Graphics/Chapter3/chap_3_sect_1_prob_13_plot.eps", onefile=FALSE, horizontal=FALSE) }
plot(ts, ys, type='l', xlab='t', ylab='y(t)', main='Question 13')
grid()
if( save_figs ){ dev.off() }


# Question 14:
#
pr = polyroot( c( -4, 1, 2 ) )
print(pr)
r1 = Re(pr[1]) # drop the 0 complex part
r2 = Re(pr[2])
A = matrix( c( 1, 1, r1, r2 ), nrow=2, ncol=2, byrow=T )
b = c( 0, 1 )
x = solve( A, b )
print( x )

ts = seq( 0, 5, length.out=100 )
ys = x[1] * exp(r1 * ts) + x[2] * exp(r2 * ts)
if( save_figs ){ postscript("../../WriteUp/Graphics/Chapter3/Chap_3_sect_1_prob_14_plot.eps", onefile=FALSE, horizontal=FALSE) }
plot(ts, ys, type='l', xlab='t', ylab='y(t)', main='Question 14')
grid()
if( save_figs ){ dev.off() }


# Question 15:
#
A = matrix( c( exp(1), exp(-9), exp(1), -9*exp(-9) ), nrow=2, ncol=2, byrow=T )
b = c( 1, 0 )
print( solve( A, b ) )


# Question 16:
#
A = matrix( c( exp(1), exp(+1), exp(-1)/2, -exp(+1)/2 ), nrow=2, ncol=2, byrow=T )
b = c( 1, -1 )
print( solve( A, b ) )