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{\bf 1.}   Solve the initial value problem: $ty' + 2y = t^3$, $\, y(2)=1$.
 

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{\bf 2.}  Find $\alpha$ so that the following equation is exact and then 
solve the equation for that value of $\alpha$: 
$$
\left( 3x^2 - \alpha xy +2 \right) dx + \left( 6y^2 - x^2 +3 \right) dy = 0.
$$


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{\bf 3.}  Solve the initial value problem: $ 2y''+y'-3y=0,\,y(0)=1,\, y'(0)=0.$
 

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{\bf 4.}  Find the general solution of:  $ y''-y'=e^{2t}.$
 

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{\bf 5a.} Find all singular points of the differential equation:
$$
x^{2}\left( 1 - x^2 \right) y'' + xy' + 3xy = 0
$$
and classify them as regular or irregular.



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{\bf 5b.} Let $f = e^{x} \cos{x}$ and $g = e^{-x}\sin{x}$. Compute 
the Wronkian $W(f,g)$.

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{\bf 6a.} Circle the differential equation whose direction field
is shown on the picture below.
$$A.\  y'=t+2y,\quad B.\  y'=-t+2y,\quad C.\  y'=-t-2y,\quad 
D.\  y'=2t-y,\quad E.\  y'=2t+y$$



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{\bf 6b.} Find the constant (equilibrium) solutions of the differential equation
$$
\frac{dy}{dt} = y^{2} \left( 4 - y^2 \right).
$$
Classify each one as asymptotically stable, unstable or semistable.

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{\bf 7.} Find a particular solution of: 
$y'' + 2y' - 3y = e^{t} + 65\cos{2t}$. 


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{\bf 8.}  Let: $$A=\pmatrix  1&-4&-1&-1&2\\
2&-8&-1&1&2\\
-1&4&2&4&-3\\3&-12&0&6&0
\endpmatrix .$$

a)  Find the reduced echelon form of $A.$

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b)  Find the rank of $A$.

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{\bf 9.}  Let: $$A=\pmatrix  1&-4&0&2&0\\
0&0&1&3&0\\
0&0&0&0&1\\0&0&0&0&0
\endpmatrix 
\;\;\;
{\text{and}} 
\;\;\;
b = \pmatrix 5\\3\\2\\0 \endpmatrix.$$

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a) Find the solution space $W$ of the homogeneous system:
$A x = 0$.

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b) Find the general solution of the inhomogeneous system:
$A x = b$.

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{\bf 10.} Solve the differential equation:
$$
y'' - xy' -2y = 0
$$ 
by means of a power series solution about the point $x_{0}=0$ as follows.

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a) Find the recurrence relation for the coefficients.

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b) Find the first 5 terms (that is, up to power $x^4$) in each of two 
linearly independent solutions.

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