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grade |
HW subject |
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1 |
93/100 |
Section 1.2,1.4,1.5. Solve separable ODE. Solve ODE by \(u=\frac{y}{t}\) substitution.
population model problem. Asking what the population will be after
some time |
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2 |
98/100 |
section 1.8,1.10,1.13. Tank mixing. Finding Orthogonal projection. Find
where solution exist. Show that some given solution for initial value
ODE is unique. Euler numerical solution problem. |
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3 |
100/100 |
section 2.1,2.2,2.4,2.5. Show that 2 functions are linearly independent.
Finding Wronskian. Solving second order ODE with constant
coefficients. Using Variation of parameters to find particular solution.
Using Guessing (undetermined coefficients) method to find particular
solution (RHS is \(1+t^2+e^{-2 t}\)). |
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4 |
100/100 |
section 2.6,2.9,2.10. Vibration problem. Using Laplace method to
solve second order initial value problem. Finding inverse Laplace of
expression. |
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5 |
100/100 |
section 3.1-3.5. Converting pair of first order ODE’s to system.
Determine if set of vectors form vector space. (check if closed under
addition or scalar multiplication). Find basis in 3D given 2 basis (i.e.
need to find third base vector). Given 3 solutions, determine if they
are linearly independent. (solve \(c_i x^i=0\) for \(c_i\) and show all \(c\) are zero. Find
determinant of 4 by 4 matrix. Finding inverse of Matrix. |
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6 |
100/100 |
section 3.8-3.9,3.10. Solving system \(x'=Ax\) using the eigenvalue/eigenvector
method, eigenvalues all different and real. Same as above, but 2 of
eigenvalues are complex. When one eigenvalue is complex, just find the
eigenvector for it, and find the real and imaginary parts of \(x(t)=e^{\lambda t} v(t)\) which will
give the two solutions associated with both complex eigenvalues. i.e.
only need to find one eigenvector with there are two complex eigenvalues
(since they are conjugates). Same as above, but one eigenvalue of
multiplicity 3. |
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7 |
94/100 |
section 4.1,4.2,4.3. Find all equilibrium points. Determine the stability
of all solutions to system (find the eigenvalues). Given non-linear
system, determine if origin is equilibrium point and check if stable or
not (Use the Jacobian). If non-linear system, and real part is zero, then
unable to decide. |
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8 |
99/100 |
section 4.4,4.7. Finding orbit equation for 2 by 2 system using \(\frac{dy}{dx}=\frac{g(x,y)}{f(x,y)}\). Drawing
phase diagrams. |
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