Sp2016 Practice Exam 3:  
Name_________________

Show polynomials etc. you enter in calculators and explain what you did. 
No imaginary numbers in final answers.
Read the instructions. I am frequently trying to save you time. 
Use the back of a page to show more work if you need to.
4 questions - 25 points each.

Solve y'''-8y=0               with y[0]=1, y'[0]=2, y''[0]=3

y[x]/.DSolve[ {y'''[x]-8 y[x]==0,y[0]==1,y'[0]==2,y''[0]==3},
y[x],x][[1]]//TraditionalForm

Solve y''+ 4 y = sin(2t)+(e^(2t))

y[t_]:=A (E^( 2 t)) + t* (B Sin[2t]+ C Cos[2t])
Collect[y''[t]+4 y[t]-(Sin[2t] + (E^( 2 t))),{(E^(2t)),Sin[2t], Cos[2t]}]

({{ Q1     ,    Q2   ,    Q3   ,    Q4   ,    Total   }, {      ,       ,       ,       ,       }})

Find the general solution of  (y^((7)))+(y^((6)))-(y^((5)))-5 (y^((4)))+4 y''+4 y' -4 y = 0

Write down the form you would use for undetermined coefficients for 
(y^((7)))+(y^((6)))-(y^((5)))-5 (y^((4)))+4 y''+4 y' -4 y = t (e^t)+ t sin(3t)+(t^2)
Do not solve for the constants.

({{Score/25}, {      }})

Solve y'''-4 y''+13y'=(t^2) +cos(3t)

y[t_]:=t*(A (t^2)+B t +C) + G Sin[3t]+ F Cos[3t]
Collect[y'''[t]-4 y''[t]+ 13 y'[t]-((t^2)+Cos[3t] ),{t,Sin[3t], Cos[3t]}]

Clear[y,t]
Expand[y[t]/.DSolve[ y'''[t]-4 y''[t]+ 13 y'[t]==((t^2)+Cos[3t] ),y[t],t][[1]]]/.C[i_]->0

({{Score/25}, {      }})

Show that (Subscript[y, 1])=(t^2) is a solution of  (t^2)y''-4 t y'+6y= 0.

Solve (t^2)y''-4 t y'+6y= (t^3).
Make sure you write down equations you are solving and explain your process and steps.

Clear[y,u,t]
TraditionalForm[y[t]/.DSolve[ (t^2)y''[t]-4 t y'[t]+ 6 y[t]==(t^3), y[t],t][[1]]]
y[t_]:= (t^2)u[t]
Simplify[(t^2)y''[t]-4 t y'[t]+ 6 y[t]==(t^3)]

({{Score/25}, {      }})

Show that (Subscript[y, 1])=cos[3 ln[t]] and (Subscript[y, 2])=sin[3 ln[t]] are solutions of  (t^2)y''+ t y'+9y= 0.

Solve (t^2)y''+ t y'+9y= Log[t].
Make sure you write down equations you are solving and explain your process and steps.

Clear[y,y1,y2,u1,u2,t]
y1[t_]:= Cos[3 Log[t]]
y2[t_]:= Sin[3 Log[t]]
Simplify[
Solve[ {
y1[t] u1'[t]+y2[t] u2'[t]==0, 
y1'[t] u1'[t]+y2'[t] u2'[t]==Log[t]/(t^2)
},
{u1'[t],u2'[t]}
]]

Expand[ Integrate[ {-((Log[t] Sin[3 Log[t]])/(3 t)),((Log[t] Cos[3 Log[t]])/(3 t))},t]]

({{Score/25}, {      }})

Solve y''+4 y'+4 y=6t (e^(-2 t)) 	with y[0]=2  and y'[0]=4

Expand[y[t]/.DSolve[{y''[t]+4 y'[t]+ 4 y[t]==6 t (E^(-2 t)),y[0]==2,y'[0]==4}, y[t],t][[1]]]

({{Score/25}, {      }})

Verify that {(x^2),1/x,1} is a Fundamental Set for the ODE (x^3)y'''[x]+2(x^2)y''[x]-2 x y'[x]==0.

Find a particular solution of
	(x^3)y'''[x]+2(x^2)y''[x]-2 x y'[x]==x+1
by substituting a guess of the form (Subscript[y, p])=A x + B ln[x]

Write down the general solution of 
(x^3)y'''[x]+2(x^2)y''[x]-2 x y'[x]==x+1

Expand[y[x]/.DSolve[{(x^3) y'''[x]+2 (x^2) y''[x]-2 x y'[x]==x+1}, y[x],x][[1]]]

({{Score/25}, {      }})