Chapter 2 First Order Equations

\(y=e^{-ax}\)

\(y=ce^{-x^3}\)

\(y=ce^{-(\ln x)^2/2}\)

\(\dst{y={c\over x^3}}\)

\(y=ce^{1/x}\)

\(\dst{y={e^{-(x-1)}\over x}}\)

\(\dst{y={e\over x\ln x}}\)

\(\dst{y={\pi\over x\sin x}}\)

\(y=2(1+x^2)\)

\(y=3x^{-k}\)

\(y=c(\cos kx)^{1/k}\)

\(\dst{y={1\over 3}+ce^{-3x}}\)

\(\dst{y={2\over x}+{c\over x}e^x}\)

\(\dst{y=e^{-x^2}\left({x^2\over2}+c\right)}\)

\(y=-\dst{e^{-x}+c\over1+x^2}\)

\(\dst{y={7\ln |x|\over x}+{3\over 2}x + {c\over x}}\)

\(y=(x-1)^{-4}(\ln |x-1| -\cos x+c)\)

\(\dst{y=e^{-x^2}\left({x^3 \over 4} + {c\over x}\right)}\)

\(\dst{y={2\ln |x|\over x^2} + {1\over 2} + {c\over x^2}}\)

\(y=(x+c)\cos x\)

\(\dst{y={c-\cos x\over (1+x)^2}}\)

\(\dst{y=- {1\over 2}{(x-2)^3\over (x-1)}+c{(x-2)^5\over (x-1)}}\)

\(y=(x+c)e^{-\sin^2x}\)

\(y=\dst{\frac{e^{x}}{x^{2}}-\frac{e^{x}}{x^{3}}+\frac{c}{x^{2}}}\).

\(y=\dst{\frac{e^{3x}-e^{-7x}}{10}}\)

\(\dst{\frac{2x+1}{(1+x^{2})^{2}}}\)

\(y=\dst{\frac{1}{x^{2}}\ln\left(\frac{1+x^{2}}{2}\right)}\)

\(\dst{y={1\over 2} (\sin x +\csc x)}\)

\(\dst{y=\frac{2\ln |x|}{x} + \frac{x}{2} - \frac{1}{2x}}\)

\(y=(x-1)^{-3} \left[\ln (1-x)-\cos x\right]\)

\(\dst{y=2x^2 + {1\over x^2} \quad (0,\infty)}\)

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

\(\dst{y={1 \over 2}+{5\over 2}e^{-x^2}}\)

\(\dst{y={\ln|x-1|+\tan x+1\over(x-1)^3}}\)

\(\dst{y={\ln|x|+x^2+1\over(x+2)^4}}\)

\(\dst{y=(x^2-1)\left(\displaystyle{{1\over2} \ln |x^2-1|-4}\right)}\)

\(\dst{y=-(x^2-5)\left(7+\ln|x^2-5|\right)}\)

\(\dst{y=e^{-x^2}\left(3 +\int^x_0 t^2e^{t^2}\, dt\right)}\)

\(\dst{y={1\over x}\left(2 +\int^x_1{\sin t\over t}\, dt\right)}\)

\(\dst{y=e^{-x} \int^x_1 {\tan t\over t}\, dt}\)

\(\dst{y={1\over 1+x^2}\left(1+\int^x_0 {e^t\over 1+t^2}\, dt\right)}\)

\(\dst{y={1\over x}\left(2e^{-(x-1)}+e^{-x} \int_1^x e^te^{t^2}\,dt\right)}\)

\(G=\dst{{r\over\lambda}+\left(G_0-{r\over\lambda}\right)e^{-\lambda t}}\)  \(\lim_{t\to\infty}G(t)=\dst{r\over\lambda}\)

(a) \(\dst{y=y_0 e^{-a(x-x_0)}+ e^{-ax} \int^x_{x_0} e^{at}f(t)\, dt}\)

(a) \(\dst{y=\tan^{-1}\left({1\over 3} + ce^{3x}\right)}\)   (b) \(y=\pm\dst{\left[\ln\left(\dst{1\over x}+\dst{c\over x^2}\right)\right]^{1/2}}\)

    (c) \(\dst{y=\exp\left(x^2 + {c\over x^2}\right)}\)   (d) \(\dst{y=-1 + {x\over c + 3\ln |x|}}\)

\(y=2\pm\sqrt{2(x^3+x^2+x+c)}\)

\(\ln (|\sin y|) = \cos x+c\);   \(y \equiv k \pi\),  \(k=\) integer

\(\dst{y={c\over x-c} \quad y \equiv -1}\)  

\(\dst{(\ln y)^2\over2}=-{x^3\over3}+c\)

\(y^3+3 \sin y + \ln |y|+\ln (1+x^2)+\tan^{-1}x=c\); \(y\equiv0\)

\(y=\pm\dst{\left(1+\left(x\over1+cx\right)^2\right)^{1/2}}\); \(y\equiv\pm1\)

\(\dst{y=\tan \left( {x^3\over3}+c\right)}\)

\(y=\dst{c\over\sqrt{1+x^2}}\)

\(\dst{y= {2 - ce^{(x-1)^2/2}\over 1-ce^{(x-1)^2/2}}; \quad y \equiv 1}\)

\(\dst{y=1+\big(3x^2+9x+c)^{1/3}}\)

\(\dst{y=2+\sqrt{{2\over3}x^3+3x^2+4x -{11\over3}}}\)

\(\dst{y={e^{-(x^2-4)/2} \over2-e^{-(x^2-4)/2}}}\)

\(y^3+2y^2+x^2+\sin x=3\)

\(\dst{(y+1)(y-1)^{-3}(y-2)^2=-256(x+1)^{-6}}\)

\(y=-1+3e^{-x^2}\)

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

\(y \equiv -1; \quad (- \infty,\infty)\)  

\(\dst{y={4-e^{-x^2}\over2-e^{-x^2}}; \quad (-\infty,\infty)}\)

\(\dst{y={-1+\sqrt{4x^2-15} \over2}; \quad\left({\sqrt{15}\over2}, \infty\right)}\)

\(y=\dst{\frac{2}{1+e^{-2x}}}\)   \((-\infty,\infty)\)

\(y=- \sqrt{25-x^2}\);   \((-5,5)\)

\(y \equiv 2, \quad (- \infty,\infty)\)

\(y=3\dst{\left(x+1\over2x-4\right)^{1/3}}\);   \((-\infty,2)\)

\(y=\dst{x+c\over1-cx}\)

\(y=-x\cos c+\sqrt{1-x^2}\sin c; \quad y \equiv 1; y \equiv -1\)

\(y=-x+3\pi/2\)

\(P=\dst{P_0\over\alpha P_0+(1-\alpha P_0)e^{-at}}\); \(\lim_{t\to\infty}P(t)=1/\alpha\)

\(I=\dst{SI_0\over I_0+(S-I_0)e^{-rSt}}\)

If \(q=rS\) then \(I=\dst{I_0\over1+rI_0t}\) and \(\lim_{t\to\infty}I(t)=0\). If \(q\ne Rs\), then     \(I=\dst{\alpha I_0\over I_0+(\alpha-I_0)e^{-r\alpha t}}\). If \(q<rs\), then \(\lim_{t\to\infty}I(t)=\alpha=S-\dst{q\over r}\)

    if \(q>rS\), then \(\lim_{t\to\infty}I(t)=0\)

\(f=ap, \quad \mbox{where \)a\(=constant}\)

\(y=e^{-x}\left(-1\pm\sqrt{2x^2+c}\right)\)

\(y=x^2\left(-1+\sqrt{x^2+c}\right)\)

\(y=e^x\left(-1+(3xe^x+c)^{1/3}\right)\)

\(y=e^{2x}(1\pm\sqrt{c-x^2})\)

(a) \(y_1=1/x\);   \(g(x)=h(x)\)

    (b) \(y_1=x\);   \(g(x)=h(x)/x^2\)   (c) \(y_1=e^{-x}\); \(g(x)=e^xh(x)\)

    (d) \(y_1=x^{-r}\);  \(g(x)=x^{r-1}h(x)\)   (e)  \(y_1=1/v(x)\);  \(g(x)=v(x)h(x)\)

(a),  (b) \(x_0\ne k \pi\) (\(k=\) integer)

(a),  (b) \((x_0,y_0)\ne(0,0)\)

(a),  (b) \(x_0y_0\ne(2k+1){\pi\over2}\) (\(k\)= integer)

(a),  (b) \(x_0y_0>0\) and \(x_0y_0\ne1\)

(a) all \((x_0,y_0)\)   (b) \((x_0,y_0)\) with \(y_0\ne 0\)  

(a), (b) all \((x_0,y_0)\)

(a), (b) all \((x_0,y_0)\)

(a), (b) \((x_0,y_0)\) such that \(x_0\ne 4y_0\)

(a) all \((x_0,y_0)\) (b) all \((x_0,y_0)\ne(0,0)\)

(a) all \((x_0,y_0)\)
(b) all \((x_0,y_0)\) with \(y_0\ne \pm 1\)

(a), (b) all \((x_0,y_0)\)

(a), (b) all \((x_0,y_0)\) such that \(x_0+y_0>0\)

(a), (b) all \((x_0,y_0)\) with \(x_0\ne 1,\quad y_0\ne(2k+1)\dst{{\pi\over2}}\) (\(k=\) integer)

\(\dst{y = \left( {3\over 5} x+1 \right)^{5/3},\; - \infty < x < \infty,}\) is a solution.

Also,

\[ y = \left\{ \begin{array}{cl} 0, & -\infty< x \le -{5\over 3} \\[3pt] \left({3\over 5}x + 1\right)^{5/3}, & -{5\over3} < x < \infty \end{array}\right. \]

is a solution, For every \(\dst{a \ge {5\over 3}}\), the following function is also a solution:

\[ y = \left\{ \begin{array}{cl} \left( {3\over 5} (x+a) \right)^{5/3}, & - \infty < x < -a, \\[3pt] 0, & -a \le x \le -{5\over 3} \\[3pt] \left({3\over 5}x + 1\right)^{5/3}, & -{5\over3} < x < \infty. \end{array}\right. \]

(a) all \((x_0,y_0)\) (b) all \((x_0,y_0)\) with \(y_0\ne1\)

\(y_1\equiv1\); \(y_2=1+|x|^3\); \(y_3=1-|x|^3\); \(y_4=1+x^3\); \(y_5=1-x^3\)

\(y_6=\dst{\left\{\begin{array}{ccl}1+x^3,&x\ge 0,\\ 1,&x<0\end{array}\right.}\);   \(y_7=\dst{\left\{\begin{array}{ccl}1-x^3,&x\ge 0,\\ 1,&x<0\end{array}\right.}\);

\(y_8=\dst{\left\{\begin{array}{ccl}1,&x\ge 0,\\ 1+x^3,&x<0\end{array}\right.}\);   \(y_9=\dst{\left\{\begin{array}{ccl}1,&x\ge 0,\\ 1-x^3,&x<0\end{array}\right.}\)

\(\dst{y=1+(x^2+4)^{3/2}, \quad - \infty < x < \infty}\)

(a) The solution is unique on \((0,\infty)\). It is given by

    (b)

is a solution of (A) on \((-\infty,\infty)\). If \(\alpha\ge0\), then

and

are also solutions of (A) on \((-\infty,\infty)\).

\(y=\dst{1\over 1-ce^x}\)

\(y=x^{2/7}(c-\ln|x|)^{1/7}\)

\(y=e^{2/x}(c-1/x)^2\)

\(y=\pm\dst{{\sqrt{2x+c}\over1+x^2}}\)

\(y=\pm(1-x^2+ce^{-x^2})^{-1/2}\)

\(y=\dst{\left[x\over3(1-x)+ce^{-x}\right]^{1/3}}\)

\(y=\dst{2\sqrt2\over\sqrt{1-4x}}\)

\(y=\dst{\left[1-{3\over2}e^{-(x^2-1)/4}\right]^{-2}}\)

\(y=\dst{1\over x(11-3x)^{1/3}}\)

\(y=(2e^x-1)^2\)

\(y=(2e^{12x}-1-12x)^{1/3}\)

\(y=\dst{\left[5x\over2(1+4x^5)\right]^{1/2}}\)

\(y=(4e^{x/2}-x-2)^2\)

\(P=\dst{P_0e^{at}\over1+aP_0\int_0^t\alpha(\tau)e^{a\tau}\,d\tau}\);  \(\lim_{t\to\infty}P(t)= \dst{\left\{\begin{array}{rl}\infty&\mbox{ if }L=0,\\0&\mbox{ if }L=\infty,\\ 1/aL&\mbox{ if } 0<L<\infty.\end{array}\right.}\)

\(y=x(\ln|x|+c)\)

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

\(y=\pm x(4\ln|x|+c)^{1/4}\)

\(y=x\sin^{-1} (\ln |x|+c)\)

\(y=x \tan(\ln |x|+c)\)

\(y=\pm x\sqrt{cx^2-1}\)

\(y=\pm x\ln(\ln|x|+c)\)

\(y=-\dst{2x\over 2 \ln |x|+1}\)

\(y=x(3 \ln x+27)^{1/3}\)

\(y=\dst{{1\over x}\left({9-x^4 \over 2}\right)^{1/2}}\)

\(y=-x\)

\(y=-\dst{x(4x-3)\over(2x-3)}\)

\(y=x\sqrt{4x^6-1}\)

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

\((x+y)\ln |x|+y(1-\ln |y|)+cx=0\)

\((y+x)^3=3x^3(\ln|x|+c)\)

\((y+x)=c(y-x)^3; \quad y =x;\quad y=-x\)

\(y^2(y-3x)=c;\quad y\equiv0;\; y=3x\)

\((x-y)^3(x+y)=cy^2x^4; \quad y=0;\; y=x;\; y=-x\)

\(\dst{{y\over x}+{y^3\over x^3}}=\ln |x|+c\)

Choose \(X_0\) and \(Y_0\) so that

\((y+2x+1)^4(2y-6x-3)=c; \quad y=3x+3/2; \quad y=-2x-1\)

\((y+x-1)(y-x-5)^3=c; \quad y=x+5; \quad y=-x+1\)

\(\ln|y-x-6|-\dst{2(x+2)\over y-x-6}=c; \quad y=x+6\)

\((y_1=x^{1/3})\) \(y=x^{1/3}(\ln|x|+c)^{1/3}\)

\(y_1=x^3\); \(y=\pm x^3\sqrt{cx^6-1}\)

\(y_1=x^2\); \(y=\dst{x^2(1+cx^4)\over1-cx^4}\; y=-x^2\)

\(y_1=e^x\); \(y=-\dst{e^x(1-2ce^x)\over1-ce^x};\quad y=-2e^x\)

\(y_1=\tan x\); \(y=\tan x\; \tan(\ln|\tan x|+c)\)

\(y_1=\ln x\); \(y=\dst{2\ln x\; \left(1+c(\ln x)^4\right)\over1-c(\ln x)^4};\quad y=-2\ln x\)

\(y_1=x^{1/2}\); \(y=x^{1/2}(-2\pm\sqrt{\ln|x|+c})\)

\(y_1=e^{x^2}\); \(y=e^{x^2}(-1\pm\sqrt{2x^2+c})\)

\(y=\dst{-3+\sqrt{1+60x}\over2x}\)

\(y=\dst{-5+\sqrt{1+48x}\over2x^2}\)

\(y=1+\dst{1\over x+1+ce^x}\)

\(y=e^x-\dst{1\over1+ce^{-x}}\)

\(y=1-\dst{1\over x(1-cx)}\)

\(y=x-\dst{2x\over x^2+c}\)

\(2x^3y^2=c\)

\(3y\sin x+2x^2e^x+3y=c\)

Not exact

\(x^2-2xy^2+4y^3=c\)

\(x+y=c\)

Not exact

\(2y^2\cos x+3xy^3-x^2=c\)

Not exact

\(x^3+x^2y+4xy^2+9y^2=c\)

Not exact

\(\ln |xy|+x^2+y^2=c\)

Not exact

\(x^2+y^2=c\)

\(x^2y^2e^x+2y+3x^2=c\)

\(x^3e^{x^2+y}-4y^3+2x^2=c\)

\(x^4e^{xy}+3xy=c\)

\(x^3\cos xy+4y^2+2x^2=c\)

\(y=\dst{x+\sqrt{2x^2+3x-1}\over x^2}\)

\(y=\sin x-\dst{\sqrt{1-{\tan x\over 2}}}\)

\(\dst{y=\left({e^x-1\over e^x+1}\right)^{1/3}}\)

\(y=1+2\tan x\)

\(y=\dst{x^2-x+6\over(x+2)(x-3)}\)

\(\dst{{7x^2\over2}+4xy+{3y^2\over 2}=c}\)

\((x^4y^2+1)e^x+y^2=c\)

(a) \(M(x,y)=2xy+f(x)\) (b) \(M(x,y)=2(\sin x+x\cos x) (y\sin y+\cos y)+f(x)\)

   (c) \(M(x,y)=ye^x-e^y\cos x+f(x)\)

(a) \(N(x,y)=\dst{x^4y\over 2}+x^2+6xy+g(y)\) (b) \(N(x,y)=\dst{x\over y}+2y\sin x+g(y)\)

   (c) \(N(x,y)=x(\sin y+y\cos y)+g(y)\)

\(B=C\)

\(B=2D,\quad E=2C\)

(a) \(2x^2+x^4y^4+y^2=c\) (b) \(x^3+3xy^2=c\) (c) \(x^3+y^2+2xy=c\)

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

\(y=x^3\dst{\left(-3(x^2+1)+\sqrt{9x^4+34x^2+21}\over2\right)}\)

\(y=-e^{-x^2}\dst{\left(2x+\sqrt{9-5x^2}\over3\right)}\).

(a) \(G(x,y)=2xy+c\) (b) \(G(x,y)=e^x\sin y+c\)

(c) \(G(x,y)=3x^2y-y^3+c\) (d) \(G(x,y)=-\sin x\sinh y+c\)

(e) \(G(x,y)=\cos x\sinh y+c\)

\(\mu(x)=1/x^2\);  \(y=cx\) and \(\mu(y)=1/y^2\); \(x=cy\)

\(\mu(x)=x^{-3/2}\);  \(x^{3/2}y=c\)

\(\mu(y)={1/y^3}\);  \(y^3e^{2x}=c\)

\(\mu(x)=e^{5x/2}\);  \(e^{5x/2}(xy+1)=c\)

\(\mu(x)=e^x\);  \(e^x(xy+y+x)=c\)

\(\mu(x)=x\);  \(x^2y^2(9x+4y)=c\)

\(\mu(y)=y^2\);  \(y^3(3x^2y+2x+1)=c\)

\(\mu(y)=ye^y\);  \(e^y(xy^3+1)=c\)

\(\mu(y)=y^2\); \(y^3(3x^4+8x^3y+y)=c\)

\(\mu(x)=xe^x\);  \(x^2y(x+1)e^x=c\)

\(\mu(x)=(x^3-1)^{-4/3}\);  \(xy(x^3-1)^{-1/3}=c\) and \(x\equiv1\)

\(\mu(y)=e^y\);  \(e^y(\sin x\cos y+y-1)=c\)

\(\mu(y)=e^{-y^2}\); \(xye^{-y^2}(x+y)=c\)

\(\dst{xy\over\sin y}=c\) and \(y=k\pi\) (\(k=\) integer)

\(\mu(x,y)=x^4y^3;\; x^5y^4\ln x=c\)

\(\mu(x,y)=1/xy;\; |x|^\alpha |y|^\beta e^{\gamma x}e^{\delta y}=c\) and \(x\equiv0\), \(y\equiv0\)

\(\mu(x,y)=x^{-2}y^{-3};\; 3x^2y^2+y=1+cxy^2\) and \(x\equiv0\), \(y\equiv0\)

\(\dst{\mu(x,y) =x^{-2}y^{-1};\; -{2\over x}+y^3+3\ln |y|=c}\) and \(x\equiv0\), \(y\equiv0\)

\(\mu(x,y)=e^{ax}e^{by};\; e^{ax}e^{by}\cos xy=c\)

\(\mu(x,y)= x^{-4}y^{-3}\) (and others)  \(xy=c\)

\(\mu(x,y)=xe^y\); \(x^2 ye^y\sin x=c\)

\(\mu(x)=1/x^2\);  \(\dst{x^3y^3\over3}-\dst{y\over x}=c\)

\(\mu(x)=x+1\);  \(y(x+1)^2(x+y)=c\)

\(\mu(x,y)=x^2y^2;\; x^3y^3(3x+2y^2)=c\)

\(\mu(x,y)=x^{-2}y^{-2}\); \(3x^2y=cxy+2\) and \(x\equiv0\), \(y\equiv0\)