2.2 Separable

2.2 Separable

Separable Equation: dy/dx = g(x) h(y)

Separate

(dy/(h(y)))=g(x)dx

Integrate

∫(1/(h(y)))dy=∫g(x)dx

Sort out the constant of integration

Only solve for an explicit solution if you need to!

Examples

Ex1: x'[t]=4 t ((x^2)+1)

⟹(dx/((x^2)+1))= 4 t dt
⟹ArcTan[x]=2(t^2)+c

Ex2: y'[x]=(y-2)cos(x) and y(1)=3.

⟹(dy/(y-2))=Cos[x] dx
⟹Log[|y-2|]=Sin[x] + c
Log[|3-2|]=Sin[1]+c
⟹c=-Sin[1]

Checking Solutions

Ex1: x'[t]=4 t ((x^2)+1)

DSolve[
x'[t]==t (((x[t])^2)+1),
x[t],
t
][[1]]

Ex2: y'[x]=(y-2)cos(x) and y(1)=3.

DSolve[{
y'[x]==(y[x]-2) Cos[x],
y[1]==3
},
y[x],
x
][[1]]