M427Kmt - x 2 ) y 00-xy +4 y = 0 about x = 0. 9. A...

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M427K Midterm Exam Name NO NOTES. NO CALCULATORS. 1. Find the general solution of ( y - x 3 ) dx + ( x + y 3 ) dy = 0 . 2. Find the general solution of y 0 + 2 t 1 + t 2 y = cot t 1 + t 2 . 3. Find the general solution of y 00 + 16 y = 3 te 2 t . 4. Find the general solution of y 00 - 2 y 0 + y = e t t 2 + 1 . 5. Note that y 1 = e x is a solution to xy 00 - (2 x + 1) y 0 + ( x + 1) y = 0 . Solve the IVP consisting of this equation and the initial conditions y (1) = 1 ,y 0 (1) = 1 . Also ﬁnd the largest interval over which the solution is valid. 6. For the autonomous equation y 0 = 2( y 2 - 1) , determine the equilibrium solutions, classify their stability, and sketch a few representative solutions. 7. Using φ 0 = 0, ﬁnd the next two Picard iterates for the IVP y 0 = - 2 t ( y - 1) ,y (0) = 0 . 8. Find the ﬁrst three nonzero terms of each of two linearly independent solutions to (2+
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Unformatted text preview: x 2 ) y 00-xy +4 y = 0 about x = 0. 9. A 1000-gallon tank initially contains 200 lbs. of salt dissolved in 600 gallons of water. A brine solution containing 2lbs. of salt per gallon is pumped into the tank at a rate of 3 gal./min. The well-mixed solution is pumped out at a rate of 1 gal./min. How much salt is in the tank when it is full? 10. Express the general solution of t 2 y 00 + ty +4 y = 0 as a linear combination of two real-valued functions, and show that they are linearly independent....
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