Math 325: Differential	Equations    Quiz 1    9/10/96

1. Find the general solution of the differential equation y''''[t] - y'''[t] == t.

Solution

L[y_] := D[y,{t,4}] - D[y,{t,3}]

Solve[r^4 - r^3==0]

Since r == 0 is repeated, the general soution of the homogenoeus equation L[y[t]]==0 is

yc[t_] = c1 + c2 t + c3 t^2 + c4 E^t

The usual guess for a particular solution Y[t] such that L[Y[t]]==t would be A*t + B. We need to multiply this guess by t^3 so that Y[t] will have no terms in common with yc[t].

Y[t_] = t^3(A*t + B);

L[Y[t]]//Expand

To get t we need:

const = Solve[{24 A - 6 B==0, -24 A  == 1}]

Y[t_] = Y[t] /. First[const]

The general solution of the non-homogeneous equation is

y[t_] = yc[t] + Y[t]

2. Use the Euler method with step size h = 0.5 to find an approximate value for phi[2] where phi[t] is a solution of the initial value problem
    y'[t] == t^3 - y[t]^3, y[1] == 2

Solution

h = 0.5; f[t_,y_] = t^3 - y^3;

EMStep[{t_,y_}] := N[{t+h,y+h*f[t,y]}];

EMStep[{1,2}]

EMStep[%]

So phi[2] is approximately 1.875.