91Ó°ÊÓ

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=x^{2} \cos y$$

An object that is initially thrown vertically upward with a speed of 2 meters/second from a height of \(h\) meters takes 10 seconds to reach the ground. Set up and solve the initial-value problem that governs the motion of the object, and determine \(h\)

Show that the given relation defines an implicit solution to the given differential equation, where \(c\) is an arbitrary constant. $$x y^{2}+2 y-x=c, \quad y^{\prime}=\frac{1-y^{2}}{2(1+x y)}$$.

This problem demonstrates the variation-ofparameters method for first-order linear differential equations. Consider the first-order linear differential equation $$y^{\prime}+p(x) y=q(x)$$ (a) Show that the general solution to the associated homogeneous equation $$y^{\prime}+p(x) y=0$$ is $$y_{H}(x)=c_{1} e^{-\int p(x) d x}$$ (b) Determine the function \(u(x)\) such that $$y(x)=u(x) e^{-\int p(x) d x}$$ is a solution to \((1.6 .15),\) and hence derive the general solution to (1.6.15).

Show that the given relation defines an implicit solution to the given differential equation, where \(c\) is an arbitrary constant. $$e^{y / x}+x y^{2}-x=c, \quad y^{\prime}=\frac{x^{2}\left(1-y^{2}\right)+y e^{y / x}}{x\left(e^{y / x}+2 x^{2} y\right)}$$.

Show that the given relation defines an implicit solution to the given differential equation, where \(c\) is an arbitrary constant. \(x^{2} y^{2}-\sin x=c, \quad y^{\prime}=\frac{\cos x-2 x y^{2}}{2 x^{2} y} .\) Determine the explicit solution that satisfies \(y(\pi)=1 / \pi\).

Determine the slope field and some representative solution curves for the given differential equation. $$\diamond y^{\prime}=\frac{1-y^{2}}{2+0.5 x^{2}}$$

The temperature of an object at time \(t\) is governed by the linear differential equation $$ \frac{d T}{d t}=-k(T-5 \cos 2 t) $$At \(t=0,\) the temperature of the object is \(0^{\circ} \mathrm{F}\) and is, at that time, increasing at a rate of \(5^{\circ} \mathrm{F} / \mathrm{min.}\) (a) Determine the value of the constant \(k\) (b) Determine the temperature of the object at time \(t\) (c) Describe the behavior of the temperature of the object for large values of \(t\)

Use the result from the previous problem to solve the given differential equation. For Problem 54 impose the given initial condition as well. $$y^{\prime}=(9 x-y)^{2}, \quad y(0)=0$$

Consider the RL circuit with \(R=3 \Omega, L=0.3 \mathrm{H},\) and \(E(t)=10 \mathrm{V} .\) If \(i(0)=3 \mathrm{A},\) determine the current in the circuit for \(t \geq 0\)