(a) Show that the change of variables y = x − 1 + w transforms the Riccati differential equation y ′ + 7 x − 1 y − 3 y 2 = 3 x − 2 ( 1.8.24 ) into the Bernoulli equation w ′ + x − 1 w = 3 w 2 . ( 1.8.25 ) (b) Solve Equation (1.8.25), and hence determine the general solution to (1.8.24).
(a) Show that the change of variables y = x − 1 + w transforms the Riccati differential equation y ′ + 7 x − 1 y − 3 y 2 = 3 x − 2 ( 1.8.24 ) into the Bernoulli equation w ′ + x − 1 w = 3 w 2 . ( 1.8.25 ) (b) Solve Equation (1.8.25), and hence determine the general solution to (1.8.24).
Solution Summary: The author proves that the change of variables y=x-1+w transforms the Riccati differential equation into Bernoulli equation.
(a) Show that the change of variables
y
=
x
−
1
+
w
transforms the Riccati differential equation
y
′
+
7
x
−
1
y
−
3
y
2
=
3
x
−
2
(
1.8.24
)
into the Bernoulli equation
w
′
+
x
−
1
w
=
3
w
2
.
(
1.8.25
)
(b) Solve Equation (1.8.25), and hence determine the general solution to (1.8.24).
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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