Chapter 1

Chemicals \(A\) and \(B\) combine in the ratio \(2: 3 .\) Initially there are \(10 \mathrm{g}\) of \(\mathrm{A}\) and \(15 \mathrm{g}\) of \(\mathrm{B}\) present, and after \(5 \min , 10 \mathrm{g}\) of \(\mathrm{C}\) has been produced. Determine the amount of C that has been produced in 30 min. How long will it take for the reaction to be \(50 \%\) complete?

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. $$\diamond y^{\prime}=\frac{x \sin x}{1+y^{2}}$$

Let \(F_{1}\) and \(F_{2}\) be two families of curves with the property that whenever a curve from the family \(F_{1}\) intersects one from the family \(F_{2},\) it does so at an angle \(a \neq \pi / 2 .\) If we know the equation of \(F_{2},\) then it can be shown (see Problem 23 in Section 1.1 ) that the differential equation for determining \(F_{1}\) is $$ \frac{d y}{d x}=\frac{m_{2}-\tan a}{1+m_{2} \tan a} $$ where \(m_{2}\) denotes the slope of the family \(F_{2}\) at the point \((x, y)\) Use Equation \((1.8 .16)\) to determine the equation of the family of curves that cuts the given family at an angle \(\alpha=\pi / 4\) $$x^{2}+y^{2}=c$$

Prove that if \(\left(M_{y}-N_{x}\right) / M=g(y),\) a function of \(y\) only, then an integrating factor for $$ M(x, y) d x+N(x, y) d y=0 $$ is \(I(y)=e^{-\int g(y) d y}\)

Chemicals A and B combine in the ratio 3:5 in producing the chemical C. If we have \(15 \mathrm{g}\) of \(\mathrm{A},\) use the law of mass action to determine the minimum amount of B required to produce 30 g of C.

Find the general solution to the given differential equation and the maximum interval on which the solution is valid. $$y^{\prime}=\sin x$$.

Determine the slope field and some representative solution curves for the given differential equation. $$\diamond y^{\prime}=3 x-y$$

Let \(F_{1}\) and \(F_{2}\) be two families of curves with the property that whenever a curve from the family \(F_{1}\) intersects one from the family \(F_{2},\) it does so at an angle \(a \neq \pi / 2 .\) If we know the equation of \(F_{2},\) then it can be shown (see Problem 23 in Section 1.1 ) that the differential equation for determining \(F_{1}\) is $$ \frac{d y}{d x}=\frac{m_{2}-\tan a}{1+m_{2} \tan a} $$ where \(m_{2}\) denotes the slope of the family \(F_{2}\) at the point \((x, y)\) Use Equation \((1.8 .16)\) to determine the equation of the family of curves that cuts the given family at an angle \(\alpha=\pi / 4\) $$y=c x^{6}$$

Consider the general first-order linear differential equation $$ \frac{d y}{d x}+p(x) y=q(x) $$ where \(p(x)\) and \(q(x)\) are continuous functions on some interval \((a, b)\) (a) Rewrite Equation \((1.9 .25)\) in differential form, and show that an integrating factor for the resulting equation is $$ I(x)=e^{\int p(x) d x} $$ \((1.9 .26)\) (b) Show that the general solution to Equation (1.9.25) can be written in the form $$ y(x)=I^{-1}\left\\{\int^{x} I(t) q(t) d t+c\right\\} $$ where \(I\) is given in Equation \((1.9 .26),\) and \(c\) is an arbitrary constant.

Find the general solution to the given differential equation and the maximum interval on which the solution is valid. $$y^{\prime}=x^{-2 / 3}$$.

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

Let \(F_{1}\) and \(F_{2}\) be two families of curves with the property that whenever a curve from the family \(F_{1}\) intersects one from the family \(F_{2},\) it does so at an angle \(a \neq \pi / 2 .\) If we know the equation of \(F_{2},\) then it can be shown (see Problem 23 in Section 1.1 ) that the differential equation for determining \(F_{1}\) is $$ \frac{d y}{d x}=\frac{m_{2}-\tan a}{1+m_{2} \tan a} $$ where \(m_{2}\) denotes the slope of the family \(F_{2}\) at the point \((x, y)\) Use Equation \((1.8 .16)\) to determine the equation of the family of curves that cuts the given family at an angle \(\alpha=\pi / 4\) $$x^{2}+y^{2}=2 c x$$

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$\left(x^{2}-1\right)\left(y^{\prime}-1\right)+2 y=0$$

Find the general solution to the given differential equation and the maximum interval on which the solution is valid. $$y^{\prime \prime}=x e^{x}$$.

Determine the slope field and some representative solution curves for the given differential equation. $$\diamond y^{\prime}=\frac{2+y^{2}}{3+0.5 x^{2}}$$

Find the general solution to the given differential equation and the maximum interval on which the solution is valid. $$y^{\prime \prime}=x^{n}, n \text { an integer. }$$

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 differential equation governing a trimolecular reaction is $$ \frac{d Q}{d t}=k(\alpha-Q)(\beta-Q)(\gamma-Q) $$ where \(k, \alpha, \beta, \gamma\) are constants. Solve this differential cquation if \(\alpha, \bar{\beta}, \gamma\) are all distinct and \(Q(0)=0.\)

Solve the given initial-value problem. $$y^{\prime}=x^{2} \ln x, \quad y(1)=2$$.

\(\diamond\) (a) Determine the slope field for the differential equation $$ y^{\prime}=x^{-1}(3 \sin x-y) $$ on the interval (0,10] (b) Plot the solution curves corresponding to each of the following initial conditions: $$ \begin{aligned} y(0.5) &=0, \quad y(1)=-1 \\ y(1) &=2, \quad y(3)=0 \end{aligned} $$ What do you conclude about the behavior as \(x \rightarrow 0^{+}\) of solutions to the differential equation? (c) Plot the solution curve corresponding to the initial condition \(y(\pi / 2)=6 / \pi .\) How does this fit in with your answer to part (b)? (d) Describe the behavior of the solution curves for large positive \(x\)

Solve the given differential equation. $$y^{\prime}-x^{-1} y=4 x^{2} y^{-1} \cos x, \quad x>0$$

Solve the given initial-value problem. $$y^{\prime \prime}=\cos x, \quad y(0)=2, \quad y^{\prime}(0)=1$$.

Solve the given differential equation. $$\frac{d y}{d x}+\frac{1}{2}(\tan x) y=2 y^{3} \sin x$$

Solve the given initial-value problem. $$y^{\prime \prime \prime}=6 x, \quad y(0)=1, \quad y^{\prime}(0)=-1, \quad y^{\prime \prime}(0)=4$$.

O Consider the differential equation $$ \frac{d i}{d t}+a i=b $$ where \(a\) and \(b\) are constants. By drawing the slope fields corresponding to various values of \(a\) and \(b,\) formulate a conjecture regarding the value of $$ \lim _{t \rightarrow \infty} i(t) $$

Solve the given differential equation. $$\frac{d y}{d x}-\frac{3}{2 x} y=6 y^{1 / 3} x^{2} \ln x$$

Solve the given initial-value problem. $$y^{\prime \prime}=x e^{x}, y(0)=3, y^{\prime}(0)=4$$.

Solve the given differential equation. $$y^{\prime}+2 x^{-1} y=6 \sqrt{1+x^{2}} \sqrt{y}, \quad x>0$$

Prove that the general solution to \(y^{\prime \prime}-y=0\) on any interval \(I\) is \(y(x)=c_{1} e^{x}+c_{2} e^{-x}\).

Solve the given differential equation. $$y^{\prime}+2 x^{-1} y=6 y^{2} x^{4}$$

A second-order differential equation together with two auxiliary conditions imposed at different values of the independent variable is called a boundary- value problem. solve the given boundary-value problem. $$y^{\prime \prime}=e^{-x}, \quad y(0)=1, \quad y(1)=0$$.

Solve the given differential equation. $$2 x\left(y^{\prime}+y^{3} x^{2}\right)+y=0$$

Determine which of the five types of differential equations we have studied the given differential equation falls into, and use an appropriate technique to find the solution to the initial-value problem. $$e^{-3 x+2 y} d x+e^{x-4 y} d y=0, \quad y(0)=0$$

The differential equation \(y^{\prime \prime}+y=0\) has the general solution \(y(x)=c_{1} \cos x+c_{2} \sin x\) (a) Show that the boundary-value problem \(y^{\prime \prime}+y=\) \(0, \quad y(0)=0, \quad y(\pi)=1\) has no solutions. (b) Show that the boundary-value problem \(y^{\prime \prime}+y=\) \(0, \quad y(0)=0, \quad y(\pi)=0\) has an infinite number of solutions.

Solve the given differential equation. $$y^{\prime}+6 x^{-1} y=3 x^{-1} y^{2 / 3} \cos x, \quad x>0$$

Determine which of the five types of differential equations we have studied the given differential equation falls into, and use an appropriate technique to find the solution to the initial-value problem. $$\left(3 x^{2}+2 x y^{2}\right) d x+\left(2 x^{2} y\right) d y=0, \quad y(1)=3$$

Verify that the given function is a solution to the given differential equation. In these problems, \(c_{1}\) and \(c_{2}\) are arbitrary constants. $$\diamond y(x)=c_{1} e^{2 x}+c_{2} e^{-3 x}, y^{\prime \prime}+y^{\prime}-6 y=0$$.

Solve the given differential equation. $$y^{\prime}+4 x y=4 x^{3} y^{1 / 2}$$

Determine which of the five types of differential equations we have studied the given differential equation falls into, and use an appropriate technique to find the solution to the initial-value problem. $$\frac{d y}{d x}-(\sin x) y=e^{-\cos x}, y(0)=\frac{1}{e}$$

Verify that the given function is a solution to the given differential equation. In these problems, \(c_{1}\) and \(c_{2}\) are arbitrary constants. $$\diamond y(x)=c_{1} x^{4}+c_{2} x^{-2}, x^{2} y^{\prime \prime}-x y^{\prime}-8 y=0, x>0$$.

Solve the given differential equation. $$\frac{d y}{d x}-\frac{1}{2 x \ln x} y=2 x y^{3}$$

Determine all values of the constants \(m\) and \(n,\) if there are any, for which the differential equation $$ \left(x^{5}+y^{m}\right) d x-x^{n} y^{3} d y=0 $$ is each of the following: (a) Exact. (b) Separable. (c) Homogeneous. (d) Linear. (e) Bernoulli.

Verify that the given function is a solution to the given differential equation. In these problems, \(c_{1}\) and \(c_{2}\) are arbitrary constants. $$\begin{aligned} &\diamond y(x)=c_{1} x^{2}+c_{2} x^{2} \ln x+\frac{1}{6} x^{2}(\ln x)^{3}\\\ &x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=x^{2} \ln x, \quad x>0 \end{aligned}$$.

Solve the given differential equation. $$\frac{d y}{d x}-\frac{1}{(\pi-1) x} y=\frac{3}{1-\pi} x y^{\pi}$$

Verify that the given function is a solution to the given differential equation. In these problems, \(c_{1}\) and \(c_{2}\) are arbitrary constants. \(\diamond y(x)=x^{a}\left[c_{1} \cos (b \ln x)+c_{2} \sin (b \ln x)\right]\) \(x^{2} y^{\prime \prime}+(1-2 a) x y^{\prime}+\left(a^{2}+b^{2}\right) y=0, x>0,\) where \(a\) and \(b\) are arbitrary constants.

Solve the given differential equation. $$2 y^{\prime}+y \cot x=8 y^{-1} \cos ^{3} x$$

Verify that the given function is a solution to the given differential equation. In these problems, \(c_{1}\) and \(c_{2}\) are arbitrary constants. $$\begin{aligned} &\diamond y(x)=c_{1} e^{x}+c_{2} e^{-x}\left(1+2 x+2 x^{2}\right)\\\ &x y^{\prime \prime}-2 y^{\prime}+(2-x) y=0, x>0 \end{aligned}$$.

Solve the given differential equation. $$(1-\sqrt{3}) y^{\prime}+y \sec x=y^{\sqrt{3}} \sec x$$

A simple nonlinear law of cooling states that the rate of change of temperature of an object is proportional to the \(square\) of the temperature difference between the object and its surrounding medium (you may assume that the temperature of the surrounding medium is constant). Set up and solve the initial-value problem that governs this cooling process if the initial temperature is \(T_{0} .\) What happens to the temperature of the object as \(t \rightarrow \infty ?\)