Difference between revisions of "Tutorial Week 1"
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===Worksheet Questions=== | ===Worksheet Questions=== | ||
| − | # For each of the following differential equations, state whether it is an ODE or a PDE, state its order (if it’s a PDE, give the order in each independent variable) and whether it is linear or nonlinear. | + | # (3 points total, 1 point each) For each of the following differential equations, state whether it is an ODE or a PDE, state its order (if it’s a PDE, give the order in each independent variable) and whether it is linear or nonlinear. |
## $t^3y''+y'=\sin(t)$ | ## $t^3y''+y'=\sin(t)$ | ||
## $\frac{\partial^2 y}{\partial x^2} + x\frac{\partial y}{\partial t} = x^4$ | ## $\frac{\partial^2 y}{\partial x^2} + x\frac{\partial y}{\partial t} = x^4$ | ||
## $e^xy+(\frac{dy}{dx})^2 = x\tan(x)$ | ## $e^xy+(\frac{dy}{dx})^2 = x\tan(x)$ | ||
| − | # Suppose that $y(t)=2-t+Ce^{5t}$ is the general solution to some differential equation. Consider the initial condition $y(0)=A$. | + | # (4 points total, 1 point each) Suppose that $y(t)=2-t+Ce^{5t}$ is the general solution to some differential equation. Consider the initial condition $y(0)=A$. |
## For what values of $A$ does $\lim_{t\to+\infty}y(t) = \infty$? | ## For what values of $A$ does $\lim_{t\to+\infty}y(t) = \infty$? | ||
## For what values of $A$ does $\lim_{t\to+\infty}y(t) = -\infty$? | ## For what values of $A$ does $\lim_{t\to+\infty}y(t) = -\infty$? | ||
## Are there any values of $A$ for which this limit is finite? | ## Are there any values of $A$ for which this limit is finite? | ||
## Sketch 5 members of this family of functions choosing values of A so that the behaviour of all types of solutions is demonstrated. | ## Sketch 5 members of this family of functions choosing values of A so that the behaviour of all types of solutions is demonstrated. | ||
| − | # Find solutions to the following ordinary differential equations. Recall that you can always check that you got the correct solution by plugging it back in to the given differential equation. Show this solution-checking process for the solution to the equation in part c. | + | # (6 points total, 2 points each) Find solutions to the following ordinary differential equations. Recall that you can always check that you got the correct solution by plugging it back in to the given differential equation. Show this solution-checking process for the solution to the equation in part c. |
## $y' - 3y=x, \quad y(0)=0$ | ## $y' - 3y=x, \quad y(0)=0$ | ||
## $xy'-2y=x^3\sin(x)$. | ## $xy'-2y=x^3\sin(x)$. | ||
Revision as of 10:38, 19 January 2015
Worksheet Questions
- (3 points total, 1 point each) For each of the following differential equations, state whether it is an ODE or a PDE, state its order (if it’s a PDE, give the order in each independent variable) and whether it is linear or nonlinear.
- $t^3y''+y'=\sin(t)$
- $\frac{\partial^2 y}{\partial x^2} + x\frac{\partial y}{\partial t} = x^4$
- $e^xy+(\frac{dy}{dx})^2 = x\tan(x)$
- (4 points total, 1 point each) Suppose that $y(t)=2-t+Ce^{5t}$ is the general solution to some differential equation. Consider the initial condition $y(0)=A$.
- For what values of $A$ does $\lim_{t\to+\infty}y(t) = \infty$?
- For what values of $A$ does $\lim_{t\to+\infty}y(t) = -\infty$?
- Are there any values of $A$ for which this limit is finite?
- Sketch 5 members of this family of functions choosing values of A so that the behaviour of all types of solutions is demonstrated.
- (6 points total, 2 points each) Find solutions to the following ordinary differential equations. Recall that you can always check that you got the correct solution by plugging it back in to the given differential equation. Show this solution-checking process for the solution to the equation in part c.
- $y' - 3y=x, \quad y(0)=0$
- $xy'-2y=x^3\sin(x)$.
- $y'=\dfrac{y}{x^2+3x+2}$.
Solutions
-
- 2nd order, linear, ODE
- 2nd order in space, 1st order in time , linear, PDE
- 1st order, nonlinear, ODE
-
- $A>2$
- $A\leq 2 $
- No
- Graphs
-
- $y = \frac{1}{9}e^{3x}-\frac{x}{3}-\frac{1}{9}$, (integrating factor)
- $y = cx^2 - x^2 cos(x)$, (integrating factor)
- $y = c(x+1)/(x+2)$, (separable equation). For the solution checking process, $y'=\frac{c}{(x+2)^2}$ and $\frac{y}{(x+1)(x+2)}=\frac{c}{(x+2)^2}$.
