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Unit 7: Differential Equations
7.7 Exponential Models with Differential Equations
Solving exponential differential equations:
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Cards (45)
In the standard form, `y` represents the dependent
variable
The typical solution of an exponential differential equation is an
exponential function
.
True
What is the general solution of the exponential differential equation
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
k
y
ky
k
y
?
y
=
y =
y
=
C
e
k
x
Ce^{kx}
C
e
k
x
The standard form of an exponential differential equation is
\frac{dy}{dx} = ky
The typical solution of an exponential differential equation is an
exponential
function.
What method is commonly used to solve exponential differential equations in their standard form?
Separation of variables
The standard form of an exponential differential equation is
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
k
y
ky
k
y
True
The constant `k` in the standard form
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
k
y
ky
k
y
represents the rate of change
The first step in solving an exponential differential equation is to understand its
form
What does `x` represent in the standard form of an exponential differential equation?
Independent variable
Steps to solve an exponential differential equation using separation of variables:
1️⃣ Rearrange the equation:
d
y
y
=
\frac{dy}{y} =
y
d
y
=
k
d
x
k\,dx
k
d
x
2️⃣ Integrate both sides:
∫
d
y
y
=
\int\frac{dy}{y} =
∫
y
d
y
=
∫
k
d
x
\int k\,dx
∫
k
d
x
3️⃣ Solve for
y
y
y
:
y
=
y =
y
=
C
e
k
x
Ce^{kx}
C
e
k
x
The relationship between \frac{dy}{dx}</latex> and `y` in the standard form is linear.
True
The constant of integration in the solution of an exponential differential equation is determined by the initial
conditions
.
What is the final step in solving for `y` after finding the antiderivatives in the separation of variables method?
Solve for `y`
What is the general solution to an exponential differential equation after applying the constant of integration?
y
=
y =
y
=
C
e
k
x
Ce^{kx}
C
e
k
x
The relationship between
d
y
d
x
\frac{dy}{dx}
d
x
d
y
and
y
y
y
in the standard form is linear.
True
Steps to solve an exponential differential equation using separation of variables
1️⃣ Separate variables: \frac{dy}{y} = k\,dx</latex>
2️⃣ Integrate both sides:
∫
d
y
y
=
\int\frac{dy}{y} =
∫
y
d
y
=
∫
k
d
x
\int k\,dx
∫
k
d
x
3️⃣ Solve for y:
y
=
y =
y
=
C
e
k
x
Ce^{kx}
C
e
k
x
The relationship between
d
y
d
x
\frac{dy}{dx}
d
x
d
y
and
y
y
y
in a general differential equation can be non-linear
Integrating both sides of
d
y
y
=
\frac{dy}{y} =
y
d
y
=
k
d
x
k \, dx
k
d
x
results in \ln |y| = kx + C</latex>.
True
What are the results of integrating
d
y
y
\frac{dy}{y}
y
d
y
and
k
d
x
k \, dx
k
d
x
separately?
ln
∣
y
∣
+
\ln |y| +
ln
∣
y
∣
+
C
1
C_{1}
C
1
and
k
x
+
kx +
k
x
+
C
2
C_{2}
C
2
Solving for
y
y
y
in \ln |y| = kx + C</latex> involves exponentiating both sides.
True
The final solution after applying the initial condition y(x_{0}) = y_{0}</latex> is
y
=
y =
y
=
y
0
e
k
(
x
−
x
0
)
y_{0}e^{k(x - x_{0})}
y
0
e
k
(
x
−
x
0
)
.
True
The solution to an exponential differential equation is
y
=
y =
y
=
C
e
k
x
Ce^{kx}
C
e
k
x
.
True
Steps to solve an exponential differential equation using separation of variables
1️⃣ Separate variables:
d
y
y
=
\frac{dy}{y} =
y
d
y
=
k
d
x
k\,dx
k
d
x
2️⃣ Integrate both sides:
∫
d
y
y
=
\int\frac{dy}{y} =
∫
y
d
y
=
∫
k
d
x
\int k\,dx
∫
k
d
x
3️⃣ Solve for y:
y
=
y =
y
=
C
e
k
x
Ce^{kx}
C
e
k
x
What is the standard form of an exponential differential equation?
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
k
y
ky
k
y
What method is used to solve exponential differential equations in standard form?
Separation of variables
A general differential equation always includes a constant term.
False
What is the new constant of integration after solving for
y
y
y
in the separation of variables method?
A
=
A =
A
=
e
C
e^{C}
e
C
What is the next step after separating variables in an exponential differential equation?
Integrate both sides
Steps to solve for
y
y
y
after integrating both sides of an exponential differential equation
1️⃣ Take the antiderivatives:
ln
∣
y
∣
=
\ln |y| =
ln
∣
y
∣
=
k
x
+
kx +
k
x
+
C
C
C
2️⃣ Solve for y by exponentiating both sides:
y
=
y =
y
=
C
e
k
x
Ce^{kx}
C
e
k
x
What does applying an initial condition allow you to determine in the exponential solution?
The constant of integration
To separate the variables in the equation
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
k
y
ky
k
y
, we rearrange it to \frac{dy}{y}
The constant of integration `C` in the solution y = Ce^{kx}</latex> is determined by
initial conditions
.
True
In the standard form of an exponential differential equation, what does `y` represent?
Dependent variable
What does `k` represent in the standard form of an exponential differential equation?
Constant
Steps to solve an exponential differential equation using separation of variables
1️⃣ Rearrange the equation:
d
y
y
=
\frac{dy}{y} =
y
d
y
=
k
d
x
k\,dx
k
d
x
2️⃣ Integrate both sides:
∫
d
y
y
=
\int\frac{dy}{y} =
∫
y
d
y
=
∫
k
d
x
\int k\,dx
∫
k
d
x
The standard form of an exponential differential equation involves a linear relationship between
d
y
d
x
\frac{dy}{dx}
d
x
d
y
and `y`.
True
After integrating both sides, the equation
ln
∣
y
∣
=
\ln |y| =
ln
∣
y
∣
=
k
x
+
kx +
k
x
+
C
C
C
includes the constant of integration.
Match the characteristics with the correct type of differential equation:
Relationship between
d
y
d
x
\frac{dy}{dx}
d
x
d
y
and `y` ↔️ Linear for standard form
Typical solution ↔️ Exponential function for standard form
The standard form of an exponential differential equation always includes a
constant
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