Question Type 1: Converting a second order differential equation into a system of first order differential equations Exercises

Convert the second order ODE $\frac{d^2x}{dt^2}=2x-\frac{dx}{dt}+t^2$ with initial conditions $x(0)=4$, $\frac{dx}{dt}(0)=2$ into an equivalent system of first order ODEs.

Continue the Euler approximation of question 2 to $t=0.2$ with the same step size $h=0.1$. Find $x(0.2)$ and $v(0.2)$.

Using the system from question 1 and Euler’s method with step size $h=0.1$, compute approximate values of $x$ and $v=dx/dt$ at $t=0.1$.

Use Euler’s method with $h=0.1$ on $\frac{d^2x}{dt^2}=-2x-\frac{dx}{dt}$, $x(0)=2$, $x'(0)=0$, to approximate $x(0.3)$ and $x'(0.3)$.

Consider the second order ODE $\frac{d^2y}{dt^2}=-3y+2\frac{dy}{dt}+t$ with $y(0)=1$, $y'(0)=0$. Use Euler’s method with $h=0.1$ to approximate $y(0.3)$ and $y'(0.3)$.

Use Euler’s method with $h=0.1$ on the system from question 1 to approximate $x(0.5)$ and $v(0.5)$.

Approximate $x(0.4)$ and $x'(0.4)$ for $\frac{d^2x}{dt^2}=x+t^2$ with $x(0)=0$, $x'(0)=0$, using Euler’s method with $h=0.2$.

Apply Euler’s method with step size $h=0.25$ to the system in question 1 and approximate $x(1.0)$ and $v(1.0)$.

Apply Euler’s method ($h=0.2$) to the ODE $\frac{d^2x}{dt^2}=t x-\frac{dx}{dt}+\sin t$ with $x(0)=0$, $x'(0)=1$, and find $x(0.4)$, $x'(0.4)$.

Using $h=0.25$, apply Euler’s method to $\frac{d^2x}{dt^2}=\sin t - x$ with $x(0)=1$, $x'(0)=0$, to find $x(0.5)$ and $x'(0.5)$.

Use Euler’s method with $h=0.5$ on the system from question 1 to approximate $x(2.5)$ and $v(2.5)$.

Apply Euler’s method with $h=0.2$ to the ODE $\frac{d^2x}{dt^2}=e^{-t}x-\frac{dx}{dt}+t$, with $x(0)=1$, $x'(0)=1$, and approximate $x(0.6)$, $x'(0.6)$.