5.18 Second order differential equations Quiz

1. Write $\frac{d^2x}{dt^2}=3x-2\frac{dx}{dt}+t$ as a coupled system using $y=\frac{dx}{dt}$.

• A. $\frac{dx}{dt}=3x-2y+t$ and $\frac{dy}{dt}=y$
• B. $\frac{dx}{dt}=y$ and $\frac{dy}{dt}=3x+2y+t$
• C. $\frac{dx}{dt}=y$ and $\frac{dy}{dt}=3x-2y+t$
• D. $\frac{dx}{dt}=-y$ and $\frac{dy}{dt}=3x-2y+t$

2. $\frac{d^2x}{dt^2}=-x$, $x(0)=1$, $\frac{dx}{dt}(0)=0$, $h=0.5$. Find $x_1$ and $y_1$.

• A. $x_1=0.5$, $y_1=0$
• B. $x_1=1$, $y_1=-0.5$
• C. $x_1=1$, $y_1=0.5$
• D. $x_1=0.5$, $y_1=-0.5$

3. What is the phase portrait of $\frac{d^2x}{dt^2}+4x=0$?

• A. Closed ellipses around the origin
• B. A spiral towards the origin
• C. A saddle point
• D. Straight lines through the origin only

4. The eigenvalues of $\frac{d^2x}{dt^2}+3\frac{dx}{dt}+2x=0$ are found from $\lambda^2+3\lambda+2=0$. What happens to $x$?

• A. It oscillates with growing amplitude
• B. It decays to zero without oscillating
• C. It oscillates with constant amplitude
• D. It grows without bound

5. What is the origin for the equation $\frac{d^2x}{dt^2}=4x$?

• A. A stable node
• B. A spiral
• C. A centre
• D. A saddle point