Chapter 4 Applications of First Order Equations

\(Q=20e^{-(t\ln2)/3200}\) g

\({2\ln10\over\ln2}\) days

\(\dst{\tau=10{\ln2\over\ln4/3}}\) minutes

\(\dst{\tau {\ln(p_0/p_1)\over\ln2}}\)

\(\dst{{t_p\over t_q}={\ln p\over\ln q}}\)

\(\dst{k={1\over t_2-t_1}\ln {Q_1\over Q_2}}\)

20 g

\(\dst{{50 \ln2\over 3}}\) yrs

\(\dst{{25\over 2}\ln2}\)%

(a) \(=20\ln3\) yr (b). \(Q_0=100000e^{-.5}\)

(a) \(Q(t)=5000-4750e^{-t/10}\) (b) 5000 lbs

\(\dst{1\over 25}\) yrs;

\(V=V_0e^{t\ln10/2}\; 4\) hours

\(\dst{{1500\ln {4\over 3}\over\ln2}} \mbox{ yrs}; \; 2^{-4/3}Q_0\)

\(W(t)=20-19e^{-t/20}\); \(\lim_{t\to\infty}W(t)=20\) ounces

\(S(t)=10(1+e^{-t/10}); \; \lim_{t\to\infty}S(t)=10\) g

10 gallons

\(V(t)=15000+10000e^{t/20}\)

\(W(t)=4\times 10^6(t+1)^2\) dollars \(t\) years from now

\(\dst{p={100\over 25-24e^{-t/2}}}\)

(a) \(\dst{P(t)=1000e^{.06t}+50{e^{.06t}-1\over e^{.06/52}-1}}\) (b) \(5.64 \times 10^{-4}\)

(a) \(P'=rP-12M\) (b) \(\dst{P={12M\over r}(1-e^{rt})+P_0e^{rt}}\) (c) \(\dst{M\approx{rP_0\over 12(1-e^{-rN})}}\)

    (d) For (i) approximate \(M=\$402.25\), exact \(M=\$402.80\)

    for (ii) approximate \(M=\$1206.05\), exact \(M=\$1206.93\).

(a) \(T(\alpha)=\dst{-{1\over r}\ln\left(1-\left(1-e^{-rN})/\alpha\right)\right)}\mbox{ years}\)

    \(S(\alpha)=\dst{{P_0\over(1-e^{-rN})}\left[rN+\alpha\ln \left(1-(1-e^{-rN})/\alpha\right)\right]}\)

(b) \(T(1.05)=13.69\) yrs, \(S(1.05)=\$3579.94\) \(T(1.10)=12.61\) yrs,

   \(S(1.10)=\$6476.63\) \(T(1.15)=11.70\) yrs, \(S(1.15)=\$8874.98\).

\(P_0=\left\{\begin{array}{cl}\dst{S_0(1-e^{(a-r)T})\over r-a} \mbox{ if }a\ne r,\\[6pt] S_0T \mbox{ if }a=r.\end{array}\right. \)

\(\approx15.15^\circ\)F

\(\dst {T= -10+110 e^{-t \ln {11\over 9}}}\)

\(\approx 24.33^\circ\)F

(a) \(91.30^\circ\)F (b) 8.99 minutes after being placed outside (c) never

(a) 12:11:32 (b) 12:47:33

\(\dst {(85/3)^\circ C}\)

\(32^\circ\)F

\(\dst {Q(t) = 40(1-e^{-3 t/40})}\)

\(\dst {Q(t) =30-20 e^{-t/10}}\)

\(\dst {K(t)=.3-.2e^{-t/20}}\)

\(Q(50)=47.5\) (pounds)

50 gallons

\(\min q_2 = q_1/\overline{c}\)

\(\dst {Q = t+300-{234 \times 10^5\over(t+300)^2}, \quad 0 \le t \le 300}\)

(a) \(\dst{Q'+{2\over25}Q=6-2e^{-t/25}}\) (b) \(Q=75-50e^{-t/25}-25e^{-2t/25}\) (c) 75

(a) \(T=T_m+(T_0-T_m)e^{-kt}+\dst{k(S_0-T_m)\over (k-k_m)}\left(e^{-k_mt}-e^{-kt}\right)\)

    (b) \(T=T_m+k(S_0-T_m)te^{-kt}+(T_0-T_m)e^{-kt}\)   (c) \(\lim_{t\to\infty}T(t)=\lim_{t\to\infty}S(t)=T_m\)

(a) \(T'=\dst{-k\left(1+{a\over a_m}\right)T +k\left(T_{m0}+{a\over a_m}T_0\right)}\)   (b) \(\dst{T={aT_0+a_mT_{m0}\over a+a_m}+{a_m(T_0-T_{m0})\over a+a_m}e^{-k(1+a/a_m)t}}\),

    \(\dst{T_m={aT_0+a_mT_{m0}\over a+a_m}+{a(T_{m0}-T_0)\over a+a_m}e^{-k(1+a/a_m)t}}\); (c) \(\lim_{t\to\infty}T(t)= \lim_{t\to\infty}T_m(t) =\dst{aT_0+a_mT_{m0}\over a+a_m}\)

\(V=\dst{a\over b}\dst{V_0\over V_0-\left(V_0-a/b \right)e^{-at}}\),   \(\lim_{t\to\infty}V(t)=a/b\)

\(c_1=c\left(1-e^{-rt/W}\right)\), \(c_2=c\dst{\left(1-e^{-rt/W}-{r\over W}te^{-rt/W}\right)}\).

(a) \(\dst{c_n=c\left(1-e^{-rt/W}\sum_{j=0}^{n-1}{1\over j!}\left(rt\over W\right)^j\right)}\) (b) \(c\) (c) \(0\)

Let \(c_\infty=\dst{c_1W_1+c_2W_2\over W_1+W_2}\), \(\alpha=\dst{c_2W_2^2-c_1W_1^2\over W_1+W_2}\), and \(\beta=\dst{W_1+W_2\over W_1W_2}\). Then:

    (a) \(c_1(t)=c_\infty+\dst{\alpha\over W_1}e^{-r\beta t}\), \(c_2(t)=c_\infty-\dst{\alpha\over W_2}e^{-r\beta t}\)

    (b) \(\lim_{t\to\infty}c_1(t)=\lim_{t\to\infty}c_2(t)=c_\infty\)

\(v=-\dst{384\over5}\left(1-e^{-5t/12}\right);\; -\dst{384\over5}\) ft/s

\(k=12; \quad v=-16(1-e^{-2t})\)

\(v=25(1-e^{-t}); \)25 ft/s

\(v=20-27e^{-t/40}\)

\(\approx 17.10\) ft

\(v=-\dst{40(13+3e^{-4t/5})\over13-3e^{-4t/5}}\); -40 ft/s

\(v=-128(1-e^{-t/4})\)

\(T=\dst{{m\over k}\ln\left(1+{v_0k\over mg} \right)}; \quad y_m=y_0+\dst{{m\over k}\left[v_0-{mg \over k}\ln\left(1+{v_0k\over mg}\right)\right]}\)

\(v=-\dst{64(1-e^{-t})\over 1+e^{-t}}\); -64 ft/s

\(v=\alpha\dst{v_0(1+e^{-\beta t})-\alpha(1-e^{-\beta t})\over\alpha(1+e^{-\beta t})-v_0(1-e^{-\beta t})}; \quad-\alpha\), where \(\alpha=\dst{\sqrt{mg\over k}}\) and \(\beta=2\dst{\sqrt{kg\over m}}\).

\(\dst{T=\sqrt{{m\over kg}} \tan^{-1}\left(v_0 \sqrt{{k\over mg}}\right)}\) \(v=-\dst{\sqrt{mg\over k}; \, {1-e^{-2\sqrt{gk\over m}\, (t-T)}\over {1+e^{-2\sqrt{gk\over m}\, (t-T)}}}}\)

\(s'=mg-\dst{as\over s+1}\); \(a_0=mg\).

(a) \(ms'=mg-f(s)\)

(a) \(v'=-9.8+v^4/81\) (b) \(v_T\approx-5.308\) m/s

(a) \(v'=-32+8\sqrt{|v|}\);  \(v_T=-16\) ft/s (b) From Exercise 4.3. 14(c), \(v_T\) is the negative

    number such that \(-32+8\sqrt{|v_T|}=0\); thus, \(v_T=-16\) ft/s.

\(\approx 6.76\) miles/s

\(\approx 1.47\) miles/s

\(\alpha=\dst{gR^2\over(y_m+R)^2}\)

\(\overline y=0\) is a stable equilibrium; trajectories are \(v^2+\dst{y^4\over4}=c\)

\(\overline y=0\) is an unstable equilibrium; trajectories are \(v^2+\dst{2y^3\over3}=c\)

\(\overline y=0\) is a stable equilibrium; trajectories are \(v^2+\dst{2|y|^3\over3}=c\)

\(\overline y=0\) is a stable equilibrium; trajectories are \(v^2-e^{-y}(y+1)=c\)

equilibria: \(0\) (stable) and \(-2,2\) (unstable); trajectories: \(2v^2-y^4+8y^2=c\);

    separatrix: \(2v^2-y^4+8y^2=16\)

equilibria: \(0\) (unstable) and \(-2,2\) (stable); trajectories: \(2v^2+y^4-8y^2=c\);

    separatrix: \(2v^2+y^4-8y^2=0\)

equilibria: \(0,-2,2\) (stable), \(-1,1\) (unstable); trajectories:

   \(6v^2+y^2(2y^4-15y^2+24)=c\); separatrix: \(6v^2+y^2(2y^4-15y^2+24)=11\)

equilibria: \(0,2\) (stable) and \(-2,1\) (unstable);

   trajectories: \(30v^2+y^2(12y^3-15y^2-80y+120)=c\);

   separatrices: \(30v^2+y^2(12y^3-15y^2-80y+120)=496\) and

   \(30v^2+y^2(12y^3-15y^2-80y+120)=37\)

No equilibria if \(a<0\); \(0\) is unstable if \(a=0\); \(\sqrt a\) is stable and

   \(-\sqrt{a}\) is unstable if \(a>0\).

*

\(0\) is a stable equilibrium if \(a\le0\); \(-\sqrt a\) and \(\sqrt a\) are stable and \(0\) is unstable if \(a>0\).

\(0\) is unstable if \(a\le0\); \(-\sqrt a\) and \(\sqrt a\) are unstable and \(0\) is stable if \(a>0\).

\(0\) is stable if \(a\le0\); \(0\) is stable and \(-\sqrt a\) and \(\sqrt a\) are unstable if \(a\le0\).

An equilibrium solution \(\overline y\) of \(y''+p(y)=0\) is unstable if there’s an \(\epsilon>0\)

such that, for every \(\delta>0\), there’s a solution of (A) with \(\sqrt{(y(0)-\overline y)^2+v^2(0)}<\delta\), but \(\sqrt{(y(t)-\overline y)^2+v^2(t)}\ge\epsilon\) for some \(t>0\).

\(\dst{y'=-{2xy\over x^2+3y^2}}\)

\(y'=-\dst{y^2\over(xy-1)}\)

\(y'=-\dst{y(x^2+y^2-2x^2\ln|xy|)\over x(x^2+y^2-2y^2\ln|xy|)}\).

\(\dst{xy'-y=-{x^{1/2}\over2}}\)

\(y'+2xy=4xe^{x^2}\)

\(xy'+y=4x^3\)

\(y'-y=\cos x-\sin x\)

\((1+x^2)y'-2xy=(1-x)^2e^x\)

\(y'g-yg'=f'g-fg'\).

\((x-x_0)y'=y-y_0\)

\(y'(y^2-x^2+1)+2xy=0\)

\(2x(y-1)y'-y^2+x^2+2y=0\)

(a) \(y=-81+18x, (9,81) \quad y=-1+2x, (1,1)\)

     (b) \(y=-121+22x, (11,121) \quad y=-1+2x, (1,1)\)

     (c) \(y=-100-20x, (-10,100) \quad y=-4-4x, (-2,4)\)

     (d) \(y=-25-10x, (-5,25) \quad y=-1-2x, (-1,1)\)

(e) \(y=\dst{5+3x\over4}\), \((-3/5,4/5)\) \(y=-\dst{5-4x\over3}\), \((4/5,-3/5)\)

(a) \(\dst{y=-{1\over2}(1+x), (1,-1); \quad y= {5\over2}+{x\over10}, (25,5)}\)

     (b) \(\dst{y={1\over4}(4+x), (4,2) \quad y =-{1\over4} (4+x), (4,-2)}\);

     (c) \(\dst{y={1\over2}(1+x), (1,1) \quad y= {7\over2}+{x\over14}, (49,7)}\)

     (d) \(\dst{y=-{1\over2}(1+x), (1,-1) \quad y= -{5\over2}-{x\over10}, (25,-5)}\)

\(y=2x^2\)

\(\dst{y={cx\over\sqrt{|x^2-1|}}}\)

\(y=y_1+c(x-x_1)\)

\(\dst{y=-{x^3\over2}-{x\over2}}\)

\(y=-x\ln |x|+cx\)

\(\dst{y=\sqrt{2x+4}}\)

\(\dst{y=\sqrt{x^2-3}}\)

\(y=kx^2\)

\((y-x)^3(y+x)=k\)

\(y^2=-x+k\)

\(\dst{y^2=-{1\over2}\ln (1+2x^2)+k}\)

\(y^2=-2x-\ln (x-1)^2+k\)

\(\dst{y=1+{\sqrt{9-x^2\over2}}; \mbox{ those with } c>0}\)

\(\dst{\tan^{-1}{y\over x}-{1\over2}\ln(x^2+y^2)=k}\)

\(\dst{{1\over2}\ln (x^2+y^2)+(\tan\alpha)\tan^{-1}{y\over x}=k}\)