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{\Large MATH 411, Spring 2001} \\[2mm]
{\LARGE Midterm} \\[2mm]
{\large Date: February 20, 2001}\\[2mm]
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1) Solve initial-value problem:
$$
y+xy'=a(1+xy), \qquad y\Big|_{x=1/a} = -a, \quad a=\mbox{constant}.
$$

2) Find general solution to the equation 
$$
y'=\sin(x-y).
$$

3) Solve the Bernoulli equation
$$
2y'\sin x + y \cos x = y^3 \sin^2 x.
$$

4) Integrate the equation 
$$
(1-x^2y) \d x + x^2(y-x) \d y=0.
$$

5) Find general solution to the equation 
$$
y''-y'=\frac{1}{e^{x}+1}.
$$

6) Discuss simple approaches for solving the linear equation
  
  a) start with definition of linear operator;
  
  b) recall what is the definition of linear equation;
  
  c) recall what is the general (complete) solution to differential equation;
  
  d) show that general solution to inhomogeneous equation can be obtained as a
  sum of general solution to homogeneous equation and particular solution to 
  inhomogeneous equation. 

\vspace{2cm}
{\tiny 
During the examination you are allowed to use books and your notes. 
Any communications are prohibited. If you have any questions, please ask the
instructor. Good luck!
}

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