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Diff Eq 3.1

 

<== Differential Equations
Number Question Answer
See the work
1.  Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear:

 1. y''-y+t^2 = 0

 2. \displaystyle rac{dy}{dt}+ty^2=0

 3. \displaystyle rac{d^3y}{dt^3} +trac{dy}{dt}+(cos^2(t))y=t^3

 4. y''-y+y^2=0

1.

 2. 

 3. 

 4.

The work
2. Find the general solution to the homogeneous differential equation
rac{d^2y}{dx^2} - 18rac{dy}{dx} + 80y = 0

The solution has the form
y = C_1 f_1(x) + C_2 f_2(x)
with

f_1(x) =
f_2(x) =

Left to your own devices, you will probably write down the correct answers, but in case you want to quibble, enter your answers so that f_1, f_2 are normalized with their value at x=0 equal to 1.
f_1(x) =e^(8x)
f_2(x) =e^(10x)
The work
3. Find the general solution to the homogeneous differential equation
rac{d^2y}{dt^2} - 7rac{dy}{dt}  = 0

The solution has the form
y = C_1 f_1(t) + C_2 f_2(t)
with f_1(t) = and f_2(t) =
Left to your own devices, you will probably write down the correct answers, but in case you want to quibble, enter your answers so that f_1, f_2 are normalized with their value at t=0 equal to 1.

f1(t)= 1


f2(t)= e^(7t)

The work

Point Cost: 3
4. Find the general solution to the homogeneous differential equation
rac{d^2y}{dx^2} - 11rac{dy}{dx} + 28y = 0

The solution has the form
y = C_1 f_1(x) + C_2 f_2(x)
with f_1(x) = and f_2(x) =
Left to your own devices, you will probably write down the correct answers, but in case you want to quibble, enter your answers so that f_1, f_2 are normalized with their value at x=0 equal to 1.

f1(x)= e^(7x)

f2(x)= e^(4x)

The work

Point Cost: 3
5. Find the solution to the boundary value problem:
rac{d^2y}{dt^2} - 12rac{dy}{dt} + 35y = 0,     \ \ \ y(0) = 4 , y(1)= 8

The solution is
y=4.61763371e^(5t)-0.6176337103e^(7t)
The work

Point Cost: 3
6. Find y as a function of t if
2500y''   - 729y  =  0
with \quad  y(0) = 2,  \quad  y'(0) = 5 .
y=
y=152/27e^(.54t)-98/27e^(-.54t)
The work

Point Cost: 3
7. Find the function y_1 of t which is the solution of
49y''   - 42y'  + 5y  =  0
with initial conditions \quad  y_1(0) = 1,  \quad  y_1'(0) = 0 .
y_1 =

Find the function y_2 of t which is the solution of
49y''   - 42y'  + 5y  =  0

with initial conditions \quad  y_2(0) = 0,  \quad  y_2'(0) = 1 .
y_2 =

Find the Wronskian
W(t) = W(y_1,y_2).

W(t) =

Remark: You should find that W is not zero and so y_1 and y_2 form a fundamental set of solutions of
49y''   - 42y'  + 5y  =  0.
y1=-1/4e^(5/7t)+5/4e^(1/7t)
y2=7/4e^(5/7t)-7/4e^(1/7t)
W(t)=e^(--42/49t)
The work

Point Cost: 3
8. Determine whether the following pairs of functions are linearly independent or not on the whole real line.

 

 1. f(t)=t^2+9t and g(t)=t^2-9t

 2. f(t) = t and g(t)=|t|

 3. f(	heta)=9\cos 3	heta and g(	heta)=36\cos^3	heta-27\cos 	heta


 

 1.

 2.

 3.

The work
9. Find the general solution to the homogeneous differential equation
rac{d^2y}{dt^2}  + 2rac{dy}{dt} + 1y  = 0

The solution can be written in the form
y = C_1 f_1(t) + C_2 f_2(t)
with f_1(t) = and f_2(t) = Left to your own devices, you will probably write down the correct answers, but in case you want to quibble, enter your answers so that the functions are normalized with f_1(0)=1 and f_2(0)=0.

f1(t)= e^(-t)

f2(t)= te^(-t)

The work

Point Cost: 3
10. Find y as a function of t if
16y''  + 104y'  + 169y  =  0,
y(0) = 9,  \quad  y'(0) = 7 .
y=


y=9e^(-3.25t)+36.25te^(-3.25t)
The work

Point Cost: 3