Work on adding problems

src/differential_equations_1.md

@@ -774,28 +774,40 @@ since we have demonstrated that $**\phi**_{k}(t)$ is a solution of the DE.
 
 # Problems
 
-1.  [:grinning:] Solve 
+1.  [:grinning:] Solve:
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         (a)  $\dot{x}(t) = t^5$
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         (b)  $\dot{x}(t) = cos(t)$
 
 2.  [:grinning:] Solve, subject to the initial condition $x(0)=\frac{1}{2}$
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         (a) $\dot{x}(t) = -x^3$
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         (b) $\dot{x}(t) = \lambda t x$
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         (c) $\dot{x}(t) = t x^{4}$
 
 3.  [:smirk:] Solve, subject to the given initial condition
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         (a) $\dot{x}(t)=-tan(x)sin(x)$, subject to $x(0)=1$. 
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         (b) $\dot{x(t)}=\frac{1}{3} x^2+9$, subject to $x(0)=3$.
 
 4. [:grinning:] Identify which of the following systems of equations is linear.
         *Note thate you do not need to solve them!*  
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         (a) $$\dot{x_1}=t x_1 -3 x_2$$
             $$\dot{x}_2 = x_1 x_2 - x_1$$
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         (b) $$\dot{x}_1 = e^{-t}x_1$$
             $$\dot{x}_2 = \sqrt{t + cos(t)-4}x_1 + \frac{sin(t)}{t^2+t-1}x_2$$
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         (c) $$x^{(2)}_1 x_1 + \dot{x}_1 = 8 x_2$$
             $$\dot{x}_2=5tx_2 + x_1$$
 
 
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+