ODE Calculator

∫∫ Separable g(y)dy = f(x)dx μ Linear 1st y' + P(x)y = Q(x) ∂ Exact M dx + N dy = 0 vⁿ Bernoulli y' + Py = Qy^n r² 2nd Order Homo ay'' + by' + cy = 0 y_p 2nd Non-Homo ay'' + by' + cy = g(x)

Independent var: Dependent var: Use ^ for powers, * for multiply

Separable ODE: dy/dx = f(x)·g(y)

Enter g(y) and f(x) separately (equation is separated as g(y)dy = f(x)dx)

g(y) =

f(x) =

Initial Value Problem (optional)

y(x₀= ) = Leave blank for general solution

Steps: None Summary Detailed Full working

Classic ODE Examples

Separable dy/dx = xy Separable dy/dx = (x²+1)/y Linear y' − y = eˣ Linear y' + (2/x)y = x² Exact 2xy dx + x²dy Exact (y+1)dx + xdy Bernoulli y' + y = y² Bernoulli y' − y/x = −y³/x 2nd Homo y″ − y = 0 (D=4) 2nd Homo y″ + 4y = 0 (oscillator) 2nd Homo y″ − 2y' + y = 0 (repeated) Non-Homo y″ − y = eˣ Non-Homo y″ + y = sin(x) Non-Homo y″ − 3y' + 2y = 6

Separable

g(y)dy=f(x)dx

Integrate both sides

Linear 1st

y'+Py=Q

Integrating factor μ=e^∫P

Exact

Mdx+Ndy=0

Potential function F(x,y)=C

Bernoulli

y'+Py=Qyⁿ

Substitution v=y^(1−n)

2nd Homo

ay''+by'+cy=0

Characteristic equation

2nd Non-Homo

ay''+by'+cy=g

y=y_h+y_p (undetermined coeff.)

Integrating Factor \(\mu=e^{\int P\,dx}\)

Linear Solution \(y=\frac{1}{\mu}\left[\int\mu Q\,dx+C\right]\)

Characteristic \(ar^2+br+c=0\)

Discriminant \(D=b^2-4ac\)

Real Distinct \(y=C_1e^{r_1x}+C_2e^{r_2x}\)

Repeated Root \(y=(C_1+C_2x)e^{rx}\)

Complex Roots \(y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)\)

Bernoulli sub \(v=y^{1-n}\to\text{linear in }v\)

Exact Test \(\partial_y M=\partial_x N\)

Wronskian \(W=y_1y_2'-y_1'y_2\)

y'ODE Calculator

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