Direction Field Visualizer

• \(y' = t - y\)
• \(y' = \sin t\)
• \(y' = y(1 - y)\)  — logistic
• \(y' = t^2 + y^2\)
• \(y' = -y + \sin t\)
• \(y' = e^{-t^2}\)
• \(y' = y/t\)
• \(y' = ty\)
• \(y' = \cos t\,\cos y\)
• \(y' = \sqrt{|y|}\)
• \(y' = y^2 - t\)
• \(y' = \tanh(ty)\)
• \(y' = -t/y\)  — circles
• \(y' = y - y^3\)
• \(y' = \dfrac{1}{1+t^2} - y\)
• \(y' = \sin y\,\cos t\)
• \(y' = \ln(|t|+1)\cdot y\)
• \(y' = t\,e^{-y}\)
• \(y' = y(2-y)(y-1)\)  — bistable
• \(y' = \sin(ty)\)
• \(y' = \dfrac{t^3 - 2y}{t}\)  — linear
• \(y' = e^{t} + y\)  — linear, exp.\ forcing
• \(y' = e^{-2t} - 3y\)  — linear decay
• \(y' = e^{2t} + y - 1\)  — linear
• \(y' = e^{t+y}\)  — separable, blow-up
• \(y' = \dfrac{1}{(1+t^2)(1+y^2)}\)  — separable
• \(y' = \dfrac{\sin t}{t^2} - \dfrac{2y}{t}\)  — \(ty' + 2y = \dfrac{\sin t}{t}\)
• \(y' = \dfrac{\cos t}{t} - \dfrac{2y}{t}\)  — \(ty' + 2y = \cos t\)

Each example loads in arc-length mode so vertical tangents are crossed and full implicit curves close up. Bounds use a 5:3 aspect ratio matching the canvas, so circles look round. Click anywhere on the field after loading.

• \(y' = -t/y\)  —  circles \(t^2+y^2=C\)
• \(y' = -t/(4y)\)  —  ellipses \(t^2/4+y^2=C\)
• \(y' = -4t/y\)  —  tall ellipses \(t^2+y^2/4=C\)
• \(y' = t/y\)  —  hyperbolas \(t^2-y^2=C\)
• \(y' = -y/t\)  —  rect.\ hyperbolas \(ty=C\)
• \(y' = -\dfrac{2t+y}{t+2y}\)  —  tilted ellipses \(t^2+ty+y^2=C\)
• \(y' = \dfrac{t(1-2(t^2+y^2))}{y(1+2(t^2+y^2))}\)  —  Bernoulli lemniscate
• \(y' = \dfrac{t(1-2t^2)}{y}\)  —  Gerono lemniscate (figure-8)
• \(y' = \dfrac{y - t^2}{y^2 - t}\)  —  folium of Descartes
• \(y' = 1/(2y)\)  —  sideways parabolas \(t=y^2+C\)
• \(y' = -\sqrt[3]{y/t}\)  —  astroid family

• \(y' = \dfrac{t - y}{t + y}\)
• \(y' = \dfrac{t^2 + 2y^2}{2ty}\)
• \(y' = \dfrac{y + 2\sqrt{ty}}{t}\)
• \(y' = \dfrac{t + y}{t - y}\)
• \(y' = \dfrac{y(t - y)}{t(t + y)}\)
• \(y' = \dfrac{t^3 + y^3}{t\,y^2}\)
• \(y' = y/t + e^{y/t}\)
• \(y' = \dfrac{2ty}{t^2 - y^2}\)  —  tangent circles
• \(y' = \dfrac{y + \sqrt{t^2 + y^2}}{t}\)

• \(y' = \sqrt{t + y + 1}\)
• \(y' = (4t + y)^2\)

• \(y' = \dfrac{5y^3 - 2ty}{t^2}\)  —  \(n=3\)
• \(y' = \dfrac{5y^4 - 2ty}{t^2}\)  —  \(n=4\)
• \(y' = \dfrac{3t^4 + y^3}{3ty^2}\)

• \(y' = -\dfrac{2t + 3y}{3t + 2y}\)  —  \(t^2+3ty+y^2=C\)
• \(y' = \dfrac{y - 4t}{6y - t}\)  —  \(2t^2-ty+3y^2=C\)
• \(y' = -\dfrac{3t^2 + 2y^2}{4ty + 6y^2}\)  —  \(t^3+2ty^2+2y^3=C\)
• \(y' = -\dfrac{2ty^2 + 3t^2}{2t^2 y + 4y^3}\)  —  \(t^3+t^2 y^2+y^4=C\)
• \(y' = -\dfrac{1 + y\,e^{ty}}{2y + t\,e^{ty}}\)  —  \(t+y^2+e^{ty}=C\)
• \(y' = -\dfrac{e^{t}\sin y + \tan y}{e^{t}\cos y + t\sec^2 y}\)  —  \(e^{t}\sin y + t\tan y=C\)
• \(y' = -\dfrac{2t + 3}{2y - 2}\)  —  circles \((t+\tfrac{3}{2})^2+(y-1)^2=C\)
• \(y' = -\dfrac{2t - y}{2y - t}\)  —  \(t^2-ty+y^2=C\)
• \(y' = \dfrac{9t^2 + y - 1}{4y - t}\)  —  \(3t^3+ty-t-2y^2=C\)
• \(y' = -\dfrac{2ty + 1}{t^2 + 2y}\)  —  \(t^2 y + t + y^2=C\)
• \(y' = -\dfrac{2ty + y^2 + 1}{t^2 + 2ty}\)  —  \(t^2 y + ty^2 + t=C\)
• \(y' = -\dfrac{t^2 + y}{t + e^{y}}\)  —  \(t^3/3 + ty + e^{y}=C\)
• \(y' = \dfrac{2t + y}{3 + 3y^2 - t}\)  —  \(t^2+ty-3y-y^3=C\)
• \(y' = -\dfrac{t + y}{t + 2y}\)  —  \(t^2/2+ty+y^2=C\)
• \(y' = \dfrac{2t + 3ty^2}{3t^2 - 2y - y^3}\)  —  \(t^2+t^2 y^2-y^2-y^4/2 = C\)
• \(y' = -\dfrac{3t^2 - 2ty + 2}{6y^2 - t^2 + 3}\)  —  \(t^3 - t^2 y + 2t + 2y^3 + 3y=C\)
• \(y' = \dfrac{t^2 + y^2}{y^2 - 1}\)  —  vertical tangents at \(y=\pm 1\)
• \(y' = \dfrac{2y + 3}{4t - y}\)  —  integrating factor

• \(y' = \dfrac{2y + \sqrt{t^2 - y^2}}{2t}\)  —  \(y/t = \sin(\tfrac{1}{2}\ln|t|+C)\)
• \(y' = \dfrac{3y^2 + 2ty}{2ty + t^2}\)  —  homogeneous
• \(y' = \dfrac{t + 2y}{2t - y}\)  —  \((2t+4y)+(2t-2y)y'=0\)
• \(y' = y/t + \sqrt{y/t}\)  —  \(t\,dy - y\,dt = \sqrt{ty}\,dt\)

Click a function name to insert it into the equation box.

Code	Notation
sqrt(x)	\(\sqrt{x}\)
cbrt(x)	\(\sqrt[3]{x}\)
abs(x)	\(|x|\)
sin(x)	\(\sin x\)
cos(x)	\(\cos x\)
tan(x)	\(\tan x\)
cot(x)	\(\cot x\)
sec(x)	\(\sec x\)
csc(x)	\(\csc x\)
asin(x)	\(\arcsin x\)
acos(x)	\(\arccos x\)
atan(x)	\(\arctan x\)
sinh(x)	\(\sinh x\)
cosh(x)	\(\cosh x\)
tanh(x)	\(\tanh x\)
exp(x)	\(e^{x}\)
log(x)	\(\ln x\)
log10(x)	\(\log_{10} x\)
pi	\(\pi\)
e	\(e\)
ceil(x)	\(\lceil x\rceil\)
floor(x)	\(\lfloor x\rfloor\)
round(x)	\(\operatorname{round}(x)\)
max(a,b)	\(\max(a,b)\)
min(a,b)	\(\min(a,b)\)