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Magazine Problem 6
Using the method of separation of variables: Write the differential equation \(d x / d t=f(x)\) as \(d x / f(x)=d t\) and integrate both sides. $$\frac{d x}{d t}=\frac{1}{x}, x(0)=1$$
Problem 6
Find all real solutions of the differential equations. $$f^{\prime}(t)-2 f(t)=e^{2 t}$$
Problem 7
Using the method of separation of variables: Write the differential equation \(d x / d t=f(x)\) as \(d x / f(x)=d t\) and integrate both sides. \(\frac{d x}{d t}=x^{2}, x(0)=1 .\) Describe the behavior of your solution as \(t\) increases.
Problem 7
Determine the stability of the system $$\frac{d \vec{x}}{d t}=\left[\begin{array}{rr} -1 & 2 \\ 3 & -4 \end{array}\right] \vec{x}$$
Problem 7
Find all real solutions of the differential equations. $$f^{\prime \prime}(t)+f^{\prime}(t)-12 f(t)=0$$
Problem 8
Using the method of separation of variables: Write the differential equation \(d x / d t=f(x)\) as \(d x / f(x)=d t\) and integrate both sides. $$\frac{d x}{d t}=\sqrt{x}, x(0)=4$$
Problem 8
Consider a system $$\frac{d \vec{x}}{d t}=A \vec{x}$$ where \(A\) is a symmetric matrix. When is the zero state a stable equilibrium solution? Give your answer in terms of the definiteness of the matrix \(A\)
Problem 8
Find all real solutions of the differential equations. $$\frac{d^{2} x}{d t^{2}}+3 \frac{d x}{d t}-10 x=0$$
Problem 9
Using the method of separation of variables: Write the differential equation \(d x / d t=f(x)\) as \(d x / f(x)=d t\) and integrate both sides. $$\frac{d x}{d t}=\sqrt{x}, x(0)=4$$
Problem 9
Consider a system $$\frac{d \vec{x}}{d t}=A \vec{x}$$ where \(A\) is a \(2 \times 2\) matrix with tr \(A<0 .\) We are told that \(A\) has no real eigenvalues. What can you say about the stability of the system?
