Solving an equation of the form $ax \equiv\;b\;\textrm{mod}\;n$  where $\operatorname{gcd}(a,n)|r$. In this case we can find all solutions. The user is asked for the two greatest.

Solving three simultaneous congruences using the Chinese Remainder Theorem:

\[\begin{eqnarray*} x\;&\equiv&\;b_1\;&\mod&\;n_1\\ x\;&\equiv&\;b_2\;&\mod&\;n_2\\x\;&\equiv&\;b_3\;&\mod&\;n_3 \end{eqnarray*} \] where $\operatorname{gcd}(n_1,n_2,n_3)=1$

Solving three simultaneous congruences using the Chinese Remainder Theorem:

\[\begin{eqnarray*} x\;&\equiv&\;b_1\;&\mod&\;n_1\\ x\;&\equiv&\;b_2\;&\mod&\;n_2\\x\;&\equiv&\;b_3\;&\mod&\;n_3 \end{eqnarray*} \] where $\operatorname{gcd}(n_1,n_2,n_3)=1$

Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''-by=0$. Distinct roots.

Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by'+cy=0$.

Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$.

Find the solution of a first order separable differential equation of the form $(a+x)y'=b+y$.

Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$.

Find the solution of a first order separable differential equation of the form $axyy'=b+y^2$.

Solve a linear programming problem, and perform sensitivity analysis.

Determine if the following describes a probability mass function.

$P(X=x) = \frac{ax+b}{c},\;\;x \in S=\{n_1,\;n_2,\;n_3,\;n_4\}\subset \mathbb{R}$.

Solve for $x$: $\displaystyle ax+b = cx+d$

Solve for $x$: $\displaystyle \frac{px+s}{ax+b} = \frac{qx+t}{cx+d}$ with $pc=qa$.

Solve for $x$: $\displaystyle \frac{s}{ax+b} = \frac{t}{cx+d}$

Differentiate the function $(a + b x)^m  e ^ {n x}$ using the product rule.

Differentiate $f(x)=x^{m}\sin(ax+b) e^{nx}$.

The answer is of the form:
$\displaystyle \frac{df}{dx}= x^{m-1}e^{nx}g(x)$ for a function $g(x)$.

Find $g(x)$.

Differentiate $ (a+bx) ^ {m} \sin(nx)$

Differentiate $f(x) = x^m(a x+b)^n$.

Differentiate $x^m\cos(ax+b)$

Given $f(x)=(x+b)^n$. Find the gradient and equation of the chord between $(a,f(a))$ and $(a+h,f(a+h))$ for randomised values of $a$, $b$ and $h$.

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+(a^2+b^2)y=0,\;y(0)=1$ and $y'(0)=c$. 

Solve for $x(t)$, $\displaystyle\frac{dx}{dt}=\frac{a}{(x+b)^n},\;x(0)=0$

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+a^2y=0,\;y(0)=c$ and $y(1)=d$.  (Equal roots example).

Solve for $x$ and $y$:  \[ \begin{eqnarray} a_1x+b_1y&=&c_1\\   a_2x+b_2y&=&c_2 \end{eqnarray} \]

The included video describes a more direct method of solving when, for example, one of the equations gives a variable directly in terms of the other variable.

Solve for $x$: $\displaystyle ax ^ 2 + bx + c=0$.

Entering the correct roots in any order is marked as correct. However, entering one correct and the other incorrect gives feedback stating that both are incorrect.

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Solve for $x$: $\displaystyle \frac{a} {bx+c} + d= s$

Solve $\displaystyle ax + b = cx + d$ for $x$.

Solve for $x$: $\displaystyle ax ^ 2 + bx + c=0$.

Solve for $x$: $\displaystyle \frac{a} {bx+c} + d= s$