WOLFRAM|DEMONSTRATIONS PROJECT

Some Examples of Ordinary Differential Equations with Missing x or y

​
xf(
′
y
)
yf(
′
y
)
choose equation
x
3
(
′
y
)
+
′
y
+1
step 1
step 2
step 3
step 4
general solution
step 5
particular solution
step 6
parameter of general solution
2
show graph
x
3
(
′
y
)
+
′
y
+1
x
3
p
+p+1
dxdp3
2
p
+1
dydpp3
2
p
+1
y∫p3
2
p
+1p
yc+
3
4
p
4
+
2
p
2
deg6a$75718
This Demonstration solves ordinary differential equations of the forms
x=f(y')
and
y=f(y')
by introducing a new variable
p=y'
.
In the first case, the procedure is as follows:
x=f(p)
,
dx=f'(p)dp
,
dy=pdx=pf'(p)dp
,
y=∫pf'(p)dp+C
.
In the second case:
y=f(p)
,
dy=f'(p)dp
,
dx=dy/p=f'(p)/pdp
,
x=∫f'(p)pdp+C
. In both cases a solution is given in parametric form:
x=g(p)
,
y=h(p)
.