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
• \(y' = -200\,(y - \cos t)\)  → stiff (auto → BDF5)
• \(y' = -300\,(y - \sin t)\)  → stiff (auto → BDF5)
• \(y' = e^{2t} + y - 1\)  → 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 \(t^2+y^2=C\) (implicit solution)
• \(y' = -t/(4y)\)  →  ellipses \(t^2/4+y^2=C\) (implicit solution)
• \(y' = -4t/y\)  →  tall ellipses \(t^2+y^2/4=C\) (implicit solution)
• \(y' = t/y\)  →  hyperbolas \(t^2-y^2=C\) (implicit solution)
• \(y' = -y/t\)  →  rect.\ hyperbolas \(ty=C\) (implicit solution)
• \(y' = -\dfrac{2t+y}{t+2y}\)  →  tilted ellipses \(t^2+ty+y^2=C\) (implicit solution)
• \(y' = \dfrac{t(1-2(t^2+y^2))}{y(1+2(t^2+y^2))}\)  →  Bernoulli lemniscate (implicit solution)
• \(y' = \dfrac{t(1-2t^2)}{y}\)  →  Gerono lemniscate (figure-8) (implicit solution)
• \(y' = \dfrac{y - t^2}{y^2 - t}\)  →  folium of Descartes (implicit solution)
• \(y' = 1/(2y)\)  →  sideways parabolas \(t=y^2+C\) (implicit solution)
• \(y' = -\sqrt[3]{y/t}\)  →  astroid family (implicit solution)

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

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

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

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

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

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)	\(\min(a,b)\)