Often a radical change in the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. Compare the solutions of the given initial-value problems. $\frac{d y}{d x}=(y-1)^{2}, \quad y(0)=1$

Often a radical change in the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. Compare the solutions of the given initial-value problems. $\frac{d y}{d x}=(y-1)^{2}-0.01, y(0)=1$

A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially the mass is released 1 ft below the equilibrium position with a downward velocity of 5 ft/s, and the subsequent motion takes place in a medium that offers a damping force numerically equal to 2 times the instantaneous velocity.

A car is skidding to a stop on a level stretch of road. Identify all the action/reaction pairs of forces between the car and the road surface. Then draw a free-body diagram for the car.

A car moves along a horizontal road. Draw a free-body diagram; be sure to include the friction of the road that opposes the forward motion of the car.

Three ice skaters, numbered 1,2, and 3, stand in a line, each with her hands on the shoulders of the skater in front. Skater 3, at the rear, pushes on skater 2. Identify all the action/reaction pairs of forces between the three skaters. Draw a free-body diagram for skater 2, in the middle. Assume the ice is frictionless.

Find y=y(x) such that y''=9y, y(0)=-4, and y'(0)=6.

Find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable , or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. $\frac{d y}{d x}=y(2-y)(4-y)$

Show that an implicit solution of

$$2 x \sin ^ { 2 } y \: d x - \left( x ^ { 2 } + 10 \right) \cos y \: d y = 0$$

is given by

$$\ln \left( x ^ { 2 } + 10 \right) \csc y = c.$$

Find the constant solutions, if any, that were lost in the solution of the differential equation.

You hold a picture motionless against a wall by pressing on it, as shown in what we discussed earlier.

Add a third force that will cause the object to remain at rest. Label the new force $F_3$. Draw the vector starting at the black dot. The location, orientation, and length of the vector will be graded. You can move the vectors $F_1$and $F_2$ to construct the required vector, but be sure to return them into their initial positions before submitting the answer.

Consider the initial-value problem $y' + e^x y = f(x), y(0) = 1$. Express the solution of the IVP for $x > 0$ as a nonelementary integral when $f(x) = 1$. What is the solution when $f(x) = 0$? When $f(x) = e^x$?

Prove the following: Theorem. Let G be a topological group with multiplication operation $m: G \times G \rightarrow G$ and identity element e. Assume $p: \tilde{G} \rightarrow G$ is a covering map. Given $\tilde{e}$ with $p(\tilde{e})=e$, there is a unique multiplication operation on $\bar{G}$ that makes it into a topological group such that $\tilde{e}$ is the identity element and p is a homomorphism. Proof Recall that, by our convention, G and $\bar{G}$ are path connected and locally path connected. (a) Let $I: G \rightarrow G$ be the map $I(g)=g^{-1}$. Show there exist unique maps $\bar{m}: \tilde{G} \times \tilde{G} \rightarrow \tilde{G}$ and $\tilde{I}: \tilde{G} \rightarrow_{-} \tilde{G}$ with $\tilde{m}(\tilde{e} \times \tilde{e})=\tilde{e}$ and $\bar{I}(\bar{e})=\bar{e}$ such that $p \circ \tilde{m}=m \circ(p \times p)$ and $p \circ \tilde{I}=I \circ p$. (c) Show the maps $\tilde{G} \rightarrow \tilde{G}$ given by $\bar{g} \rightarrow \tilde{m}(\tilde{g} \times \tilde{I}(\tilde{g}))$ and $\tilde{g} \rightarrow \tilde{m}(\bar{I}(\tilde{g}) \times \tilde{g})$map $\tilde{G}$ to $\bar{e}$. (d) Show the maps $\bar{G} \times \tilde{G} \times \bar{G} \rightarrow \bar{G}$ given by

$$\begin{array}{l}{\tilde{g} \times \tilde{g}^{\prime} \times \tilde{g}^{\prime \prime} \rightarrow \vec{m}\left(\tilde{g} \times \bar{m}\left(\tilde{g}^{\prime} \times \tilde{g}^{\prime \prime}\right)\right)} \\ {\tilde{g} \times \tilde{g}^{\prime} \times \tilde{g}^{\prime \prime} \rightarrow \bar{m}\left(\tilde{m}\left(\tilde{g} \times \tilde{g}^{\prime}\right) \times \tilde{g}^{\prime \prime}\right)}\end{array}$$

are equal. (e) Complete the proof.

A student builds a rocket-propelled cart for a science project. Its acceleration is not quite high enough to win a prize, so he uses a larger rocket engine that provides $33\%$ more thrust, although doing so increases the mass of the cart by $12 \%$. By what percentage does the cart's acceleration increase?