1. Math 4B Practice Midterm Problems
Solutions
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2. Consider the following differential equation:
a) Find the general solution.
b) Find the solution with the initial condition y(0)=0.
General solution
Solution to IVP
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3. Consider the following differential equation:
c) Find any constant solutions and classify them as stable or unstable.
Constant solutions happen when y=constant and y’=0
y=1 is a constant solution
To see if it is stable, check the sign of the derivative on either side:
This is negative when t>0, so the solution curve will drop toward y=1
This is positive when t>0. so the solution curve will rise toward y=1
This is a STABLE solution
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4. Consider the following differential equation:
Here is a slope field for this equation. It shows the stable equilibrium at y=1 as well as the solution to our initial value problem.
y(0)=0 solution
y=1
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5. 2. Use the change of variable to solve the differential equation
Substitute into the equation:
This equation is separable, so separate and integrate…
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6. Substitute into the diff. eq.:
3) Use the change of variable z = ln(y) to solve the non-linear differential equation
Substitute into the diff. eq.:
Solving by separation. Could also use superposition or undetermined coefficients.
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7. 4) Consider the following autonomous differential equation:
a) Sketch a slope field for this equation.
b) Find any equilibrium solutions and classify them as stable or unstable.
c) Find an explicit formula for the solution that passes through the initial value y(0) = 1.
a) The slope field shows the equilibrium solutions, and their stability. We will calculate them algebraically as well.
b) To find equilibrium solutions, set y’=0:
Equilibrium solutions are the horizontal lines y=2 and y=-1
To assess the stability, plug in values on either side to find if the slope is positive or negative.
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8. 4) Consider the following autonomous differential equation:
c) Find an explicit formula for the solution that passes through the initial value y(0) = 1.
We will solve this by separation:
Use partial fractions to simplify the integral:
drop the abs value bars because the arbitrary constant C will take care of any negative signs.
Use the initial value to find C
This is an implicit solution – do some algebra and solve for y to get an explicit solution
This solution is valid for -1<y<2
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9. 5. Find the general solution to the following differential equation
We can use an integrating factor for this one.
Integrate by parts
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10. For the homogeneous solution, solve
6. An object dropped from an airplane falls under the influence of gravity and air resistance. If the force of air resistance is assumed to be proportional to the speed of the falling object, the following differential equation for the velocity v(t) is obtained. m is the mass of the object, and g and b are positive constants.
This is first order, linear and autonomous. Lots of solution methods available.
There is a constant solution when v=mg/b. This is the particular solution.
For the homogeneous solution, solve
General solution is the sum:
Maximum speed will be the steady-state solution, or the limit as t approaches ∞:
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11. To solve this, we can integrate both pieces and compare:
7. Find a value for b that makes the following equation EXACT, then find an implicit solution for the DE.
The D.E. is exact if
This D.E. is exact
To solve this, we can integrate both pieces and compare:
Implicit solution is:
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12. 8. Use the Cauchy-Euler polygon method to find an approximate value for y(3) for the following differential equation. Use a step size of 1 and show all intermediate calculations. t y y’ 1 2 3 4
Our estimate is y(3)=4.
In fact this is an exact value. It turns out that the solution to this D.E. is y(t)=t+1.
This is exactly the function we are tracing out with the Euler method (as seen by the constant slope when we calculate y’).
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13. 9. Find solutions to the following differential equations:b) c)
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14. Solve the 2x2 system of equations for c1 and c2.
9. Find solutions to the following differential equations: a)
Solve the 2x2 system of equations for c1 and c2.
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15. 9. Find solutions to the following differential equations:b)
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16. 9. Find solutions to the following differential equations:c)
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17. This is undefined when t=3 or t=-3 so we have to avoid those points.
10. For the following initial value problems, specify the region over which you can expect a unique solution to exist. You do NOT have to solve the equation. a) b)
This is undefined when t=3 or t=-3 so we have to avoid those points.
since our initial value uses t=2, our region is -3<t<3 (y can be any real #)
This is undefined when t=-4 or t=1 so we have to avoid those points.
since our initial value uses t=0, our region is -4<t<1 (y can be any real #)
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18. 11. Find the Wronskian of the following sets of functions.b) c)
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