Examples of Particular Solution

More Examples of Particular Solutions
and
The Speed Bump
EDU» dsolve('D2y +2*Dy - 3*y = 2*x*exp(x)*sin(x)
+exp(x)*cos(x)','x')
ans =
220/289*exp(x)*sin(x)-8/17*exp(x)*x*cos(x)21/289*exp(x)*cos(x)-2/17*x*exp(x)*sin(x)
+C1*exp(x)+C2*exp(-3*x)
EDU» dsolve('D2y +2*Dy - 3*y = x*x*exp(x)
+ 3*x*exp(x)','x')
ans =
1/12*exp(x)*x^3+5/16*x^2*exp(x)5/32*x*exp(x)+5/128*exp(x)
+C1*exp(x)+C2*exp(-3*x)
EDU» dsolve('D2y +2*Dy - 3*y = tan(x)','x')
ans =
1/3*i+1/2*i*exp(x)*int(1/exp(x)/(exp(2*i*x)+1),x)1/2*i*exp(-3*x)*int(exp(3*x)/(exp(2*i*x)+1),x)
+C1*exp(x)+C2*exp(-3*x)
EDU» dsolve('D2y +2*Dy - 3*y = tan(x)',
'Dy(0) = 0','y(0) = 0','x')
ans =
[ 1/3*i+1/2*i*exp(x)*int(1/exp(u)/(exp(2*i*u)+1),u = 0 ..
x)-1/2*i*exp(-3*x)*int(exp(3*u)/(exp(2*i*u)+1),u = 0 ..
x)-1/4*i*exp(x)-1/12*i*exp(-3*x)]
[
0 .. x)-1/2*i*exp(3*x)*int(exp(3*u)/(exp(2*i*u)+1),u = 0 .. x)1/4*i*exp(x)-1/12*i*exp(-3*x)]
[
0 .. x)-1/4*i*exp(x)-1/12*i*exp(-3*x)]
dsolve also fails !!!!!
Application to Speed Bump
EDU» dsolve('D2y + y = cos(v*t)',
'y(-pi/(2*v)) = 0','Dy(-pi/(2*v)) = 0','t')
ans =
1/2*(-cos(t)*cos((1+v)*t)+cos(t)*cos((1+v)*t)*vcos(t)*cos((-1+v)*t)-cos(t)*cos((-1+v)*t)*v+sin(t)*sin((1+v)*t)+sin(t)*sin((-1+v)*t)*vsin(t)*sin((1+v)*t)+sin(t)*sin((1+v)*t)*v2*v*sin(1/2*pi/v)/(-1+v^2)*cos(t)+2*v^3*sin(1/2*pi/v)/(1+v^2)*cos(t)-2*v/(1+v^2)*cos(1/2*pi/v)*sin(t)+2*v^3/(1+v^2)*cos(1/2*pi/v)*sin(t))/(-1+v^2)
EDU» syms v
EDU» whos
Name Size
ans
sol
v
1x1
1x1
1x1
Bytes Class
596 sym object
596 sym object
126 sym object
Grand total is 476 elements using 1318 bytes
EDU» sol_73 = subs(sol,v, .73)
sol_73 =
50/173*cos(t)*cos(173/100*t)+50/27*cos(t)*cos(27/100*t)-50/27*sin(t)*sin(27/100*t)+50/173*sin(t)*sin(173/100*t)7300/4671*sin(50/73*pi)*cos(t)7300/4671*cos(50/73*pi)*sin(t)
EDU» simplify(sol_73)
ans =
50/173*cos(t)*cos(173/100*t)+50/27*cos(t)*cos(27/100*t
)+50/27*sin(t)*sin(27/100*t)+50/173*sin(t)*sin(173/100*
t)7300/4671*sin(23/73*pi)*cos(t)+7300/4671*cos(23/73*pi
)*sin(t)
EDU» ezplot(sol_73,[-pi/(2*.73) pi/(2*.73)])
50/173*cos(t)*cos(173/100*t)+50/2 ~~~ t)-7300/4671*cos(50/73*pi)*sin(t)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-2
-1.5
-1
-0.5
0
t
0.5
1
1.5
2
EDU» ezplot('(-2*v)/(1-v*v)*cos(pi/(2*v))',[ 0 2])
(-2*v)/(1-v*v)*cos(pi/(2*v))
1.5
1
0.5
0
-0.5
0
0.2
0.4
0.6
0.8
1
v
1.2
1.4
1.6
1.8
2