Analytic solutions to differential equations

Examples:

<h3>A ball is thrown vertically upward</h3>

<p>A ball is thrown vertically upward with an initial velocity of $v_0=10$ m/s. There is a velocity dependent drag force directed in the opposite direction to the velocity. The total force on the ball is gravity plus the drag force $F=-mg-bv_x$, where $F$ is the force, $m$ is the mass of the ball, $g=9.81$ m/s² is the acceleration of gravity at the earth's surface, $b$ is the drag force constant, and $v_x$ is the velocity. The motion is in a line which we can take to be the $x$-axis. The differential equation that describes this motion is $m\frac{d^2x}{dt^2}+b\frac{dx}{dt}=-mg$. The equations are loaded into the analytic second order differential equation solver below. </p>

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<p>For long times, the ball falls with a constant terminal velocity $v_{\text{terminal}}=-mg/b=$ <span id="termv">-24.5</span> m/s.</p>

<h3>Damped mass-spring system</h3>

<p>We have seen that sometimes a damped mass-spring system has oscillating solutions and sometimes it has exponentially decaying solutions. To find analytic solutions assume solution there is a solution of the form $x(t)=Ce^{\lambda t}$ where $C$ and $\lambda$ are constants. This solution can be put in the differential equation and we can take the derivatives to get an algebraic equation for $\lambda$. If $\lambda$ is real, the solution decays exponentially without oscillations. However, if $\lambda$ is complex, the solutions will be $x(t)=Ce^{\text{Re}(\lambda) t+i\text{Im}(\lambda)t}$. Using Euler's equation this can be written as $x(t)=Ce^{\text{Re}(\lambda) t}(\cos(\text{Im}(\lambda)t)+i\sin(\text{Im}(\lambda)t))$. There will be oscillations if $\lambda$ is complex. The following finds solutions to the differential equation. By adjusting the parameters $m$, $b$, and $k$; undamped, underdamped, critically damped, and overdamped solutions can be found.</p>

<p>A mass $m$ is attached to a linear spring with a spring constant $k$. The spring is stretched from its equilibrium position and the mass is released from rest. A drag force acts on the mass that is in the opposite direction as the velocity $F_{\text{drag}}=-bv_x$ where $b$ is the drag force constant. The motion is in a line which we can take to be the $x$-axis. The differential equation that describes this motion is $m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0$. The equations are loaded into the analytic second order differential equation solver below. </p>

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<p style="text-align:center"><span id="termv">The period of the oscillations is $T=2\pi/\sqrt{\frac{k}{m}-\frac{b^2}{4m}}=$ 6.66 s.</span></p>






	Physik für Geodäsie 511.018 / Physik M 513.805

Analytic solutions to differential equations

Linear 2nd order differential equation solver