Math 555 Differential Equations I |
1. Solve the initial value problem
\[\begin{cases}
\tfrac{dy}{dt} = 2y - 5, \\
y(0) = y_0.
\end{cases}\]
2. Sketch the slope field for the differential equation \(y' = -y(5-y)\)
and draw three integral curves with distinctly different behavior.
3. Consider the initial value problem
\[\begin{cases}
ty' + 2y = t^2 - t + 1, \\
y(1) = \tfrac{1}{2}.
\end{cases}\]
Use the Fundamental Existence and Uniqueness Theorem to verify that a solution
exists for this initial data. Then find the solution.
4. Use the method of variation of parameters to find the general
solution of the differential equation
\[\begin{cases}
ty' + 2y = \sin t, \\
t > 0.
\end{cases}\]
Recall that to use this method you should first solve the DE \(ty' + 2y = 0\),
call this solution \(y_h\), then assume the solution of the given DE is
\(y = A(t)y_h(t)\), where \(A\) is an unknown function. Plug this back into the
original DE and solve for \(A\).
5. Use the Fundamental Existence and Uniqueness Theorem to verify that a
solution of the initial value problem exists, then solve the IVP.
\[\begin{cases}
y' = \dfrac{1-2x}{y}, \\[0.75 ex]
y(1) = -2.
\end{cases}\]
6. Consider the differential equation
\[ \dfrac{dy}{dx} = \dfrac{x^2 + xy + y^2}{x^2}. \]
Notice that this equation can be made separable by making the change of variables
\(v(x) = y(x)/x\), or \(y = x v(x)\). Use this change of variables to determine
the general solution of the DE.
7. Use the Fundamental Existence and Uniqueness Theorem to check that the
initial value problem has a unique solution, then find the solution.
\[\begin{cases}
(9x^2 + y - 1) - (4y - x)y' = 0, \\
y(1) = 0.
\end{cases}\]
8. Show the the differential equation is not exact as written, but is
exact when multiplied by the given integrating factor. Then use the integrating
factor to find the general solution of the DE.
\[\begin{cases}
(x+2)\sin y + (x\cos y)y' = 0, \\
\mu(x,y) = xe^x.
\end{cases}\]
9. Use the Fundamental Existence and Uniqueness Theorem to determine the
domain of the solution to the initial value problem without actually solving
the DE.
\[\begin{cases}
(4 - t^2)y' + 2ty = 3t^2, \\
y(1) = -3.
\end{cases}\]
10. Using Picard's Method of Successive Iterations, find a formula for the
\(n^{\mathrm{th}}\) approximation \(\varphi_n\). Then find the particular solution
\(\varphi(t) = \displaystyle \lim_{n \to \infty} \varphi_n(t)\).
\[\begin{cases}
y' = -\tfrac{y}{2} + t, \\
y(0) = 0.
\end{cases}\]
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