Add 3 simple practice problems.

src/differential_equations_1.md

@@ -5,12 +5,12 @@ title: Differential Equations
 # Differential equations
 
 A differential equation  or DE is any equation which involves both a function and some 
-derivative of that function. In this course we will be focusing on 
-*Ordinary Differential Equations*, meaning that our equations will involve 
+derivative of that function. In this lecture we will be focusing on 
+*Ordinary Differential Equations* (ODEs), meaning that our equations will involve 
 functions of one independent variable and hence any derivatives will be full 
 derivatives. Equations which involve a function of several independent variables
-and their partial derivatives are handled in courses on 
-*Partial Differential Equations*. 
+and their partial derivatives are called *Partial Differential Equations* (PDEs). and will 
+be introduced in the follow up lecture. 
 
 We consider functions $x(t)$ and define $\dot{x}(t)=\frac{dx}{dt}$, 
 $x^{(n)}(t)=\frac{d^{n}x}{dt^{n}}$. An $n$*-th* order differential equation is 
@@ -55,13 +55,13 @@ that unknowns resulting from integration may be determined.
     2. If the largest derivative in a differential equation is of first order, 
         i.e. $n=1$, then the equation is called a first order differential 
         equation.
-    3. Often you will see differential equation presented using $y(x)$ 
+    3. Often you will see differential equations presented using $y(x)$ 
         instead of $x(t)$. This is just a different nomenclature. 
             
 In this course we will be focusing on *Linear Differential Equations*, meaning 
 that we consider differential equations $x^{(n)}(t) = f(x^{(n-1)}(t), \cdots, x(t), t)$
 where the function $f$ is a linear ploynomial function of the unknown function
-$x(t)$. A simple way to spot a non-linear differential euation is to look for 
+$x(t)$. A simple way to spot a non-linear differential equation is to look for 
 non-linear terms, such as $x(t)*\dot{x}(t)$ or $x^{(n)}(t)*x^{(2)}(t)$. 
 
 Often, we will be dealing with several coupled differential equations. In this 
@@ -772,6 +772,17 @@ $(**A**- \lambda \mathbbm{1})**v**_{i} = **v**_{i-1}$. This completes the proof
 since we have demonstrated that $**\phi**_{k}(t)$ is a solution of the DE.
 
 
+# Problems
+
+1.  [:grinning:]  Solve the equation $\dot{x}(t)=5x+3$ subject to the initial 
+        condition $x(0)=\frac{2}{5}$.
+
+2.  [:smirk:] Solve the equation $\dot{x}(t)=-tan(x)sin(x)$ subject to the initial 
+        condition $x(0)=1$. 
+
+3.  [:smirk:] Solve the equation $\dot{x(t)}=\frac{1}{3} x^2+9$ subject to the 
+        initial condition $x(0)=3$.
+