diff --git a/src/differential_equations_1.md b/src/differential_equations_1.md index 3246520725f12fce3e3266a8b6f820cf77597c10..f563d07ee1ca3cbfda3df44a1e44b08b765ba7ca 100644 --- a/src/differential_equations_1.md +++ b/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: + (a) $\dot{x}(t) = t^5$ + (b) $\dot{x}(t) = cos(t)$ 2. [:grinning:] Solve, subject to the initial condition $x(0)=\frac{1}{2}$ + (a) $\dot{x}(t) = -x^3$ + (b) $\dot{x}(t) = \lambda t x$ + (c) $\dot{x}(t) = t x^{4}$ 3. [:smirk:] Solve, subject to the given initial condition + (a) $\dot{x}(t)=-tan(x)sin(x)$, subject to $x(0)=1$. + (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!* + (a) $$\dot{x_1}=t x_1 -3 x_2$$ $$\dot{x}_2 = x_1 x_2 - x_1$$ + (b) $$\dot{x}_1 = e^{-t}x_1$$ $$\dot{x}_2 = \sqrt{t + cos(t)-4}x_1 + \frac{sin(t)}{t^2+t-1}x_2$$ + (c) $$x^{(2)}_1 x_1 + \dot{x}_1 = 8 x_2$$ $$\dot{x}_2=5tx_2 + x_1$$ + +