0.4 Exercises
Introduction to Differential Equations
Your Turn
Let y ″ + 2 y ′ − 8 y = 0 y'' + 2y' - 8y = 0 y = e r x y = e^{rx} r r r r r r
Your Turn
Verify that x = C e − 2 t x = C e^{-2t} x ′ = − 2 x x' = -2x C C x ( 0 ) = 100 x(0) = 100
Your Turn
Verify that x = C 1 e − t + C 2 e 2 t x = C_1 e^{-t} + C_2 e^{2t} x ″ − x ′ − 2 x = 0 x'' - x' -2 x = 0 C 1 C_1 C 2 C_2 x ( 0 ) = 10 x(0) = 10 x ′ ( 0 ) = 0 x'(0) = 0
Your Turn
Find a solution to ( x ′ ) 2 + x 2 = 4 {(x')}^2 + x^2 = 4
Your Turn
Solve:
d
A
d
t
=
−
10
A
,
A
(
0
)
=
5
\dfrac{dA}{dt} = -10 A, \quad A(0)=5
d
H
d
x
=
3
H
,
H
(
0
)
=
1
\dfrac{dH}{dx} = 3 H, \quad H(0)=1
d
2
y
d
x
2
=
4
y
,
y
(
0
)
=
0
,
y
′
(
0
)
=
1
\dfrac{d^2y}{dx^2} = 4 y, \quad y(0)=0, \quad y'(0)=1
d
2
x
d
y
2
=
−
9
x
,
x
(
0
)
=
1
,
x
′
(
0
)
=
0
\dfrac{d^2x}{dy^2} = -9 x, \quad x(0)=1, \quad x'(0)=0
Your Turn
Is there a solution to y ′ = y y' = y y ( 0 ) = y ( 1 ) y(0) = y(1)
Your Turn
The population of city X X 100 100 20 20 X X 120 120 10 10
Your Turn
Suppose that a football coach gets a salary of one million dollars now, and a raise of 10 % 10\% s s t t
What is s ( 0 ) s(0) s ( 1 ) s(1) Approximately how many years will it take for the salary to be 10 10 Approximately how many years will it take for the salary to be 20 20 Approximately how many years will it take for the salary to be 30 30 Your Turn
Show that x = e − 2 t x = e^{-2t} x ″ + 4 x ′ + 4 x = 0 x'' + 4x' + 4x = 0
Answer
Compute x ′ = − 2 e − 2 t x'=-2e^{-2t} x ″ = 4 e − 2 t x''=4e^{-2t} ( 4 e − 2 t ) + 4 ( − 2 e − 2 t ) + 4 ( e − 2 t ) = 0 (4e^{-2t})+4(-2e^{-2t})+4(e^{-2t})=0
Your Turn
Let x y ″ − y ′ = 0 xy'' - y' = 0 y = x r y = x^r r r r r
Answer
y = x r y=x^{r} r = 0 r=0 r = 2 r=2
Your Turn
Verify that x = C 1 e t + C 2 x=C_1e^t+C_2 x ″ − x ′ = 0 x''-x' = 0 C 1 C_1 C 2 C_2 x x x ( 0 ) = 10 x(0) = 10 x ′ ( 0 ) = 100 x'(0) = 100
Answer
C 1 = 100 C_{1}=100 C 2 = − 90 C_{2}=-90
Your Turn
Solve d φ d s = 8 φ \frac{d\varphi}{ds} = 8 \varphi φ ( 0 ) = − 9 \varphi(0) = -9
Answer
φ
=
−
9
e
8
s
\varphi =-9e^{8s}
Your Turn
Solve:
d
x
d
t
=
−
4
x
,
x
(
0
)
=
9
\dfrac{dx}{dt} = -4x, \quad x(0)=9
d
2
x
d
t
2
=
−
4
x
,
x
(
0
)
=
1
,
x
′
(
0
)
=
2
\dfrac{d^2x}{dt^2} = -4x, \quad x(0)=1, \quad x'(0)=2
d
p
d
q
=
3
p
,
p
(
0
)
=
4
\dfrac{dp}{dq} = 3 p, \quad p(0)=4
d
2
T
d
x
2
=
4
T
,
T
(
0
)
=
0
,
T
′
(
0
)
=
6
\dfrac{d^2T}{dx^2} = 4 T, \quad T(0)=0, \quad T'(0)=6
Answer
x
=
9
e
−
4
t
x=9e^{-4t}
x
=
cos
(
2
t
)
+
sin
(
2
t
)
x=\cos (2t)+\sin (2t)
p
=
4
e
3
q
p=4e^{3q}
T
=
3
sinh
(
2
x
)
T=3\sinh (2x)
Classification of Differential Equations
Your Turn
Classify the following equations. Are they ODE or PDE? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant coefficient? If it is an ODE, is it autonomous?
sin
(
t
)
d
2
x
d
t
2
+
cos
(
t
)
x
=
t
2
\displaystyle \sin(t) \frac{d^2 x}{dt^2} + \cos(t) x = t^2
∂
u
∂
x
+
3
∂
u
∂
y
=
x
y
\displaystyle \frac{\partial u}{\partial x} + 3 \frac{\partial u}{\partial y} = xy
y
″
+
3
y
+
5
x
=
0
,
x
″
+
x
−
y
=
0
\displaystyle y''+3y+5x=0, \quad x''+x-y=0
∂
2
u
∂
t
2
+
u
∂
2
u
∂
s
2
=
0
\displaystyle \frac{\partial^2 u}{\partial t^2} + u\frac{\partial^2 u}{\partial s^2} = 0
x
″
+
t
x
2
=
t
\displaystyle x''+tx^2=t
d
4
x
d
t
4
=
0
\displaystyle \frac{d^4 x}{dt^4} = 0
Your Turn
If u → = ( u 1 , u 2 , u 3 ) \vec{u} = (u_1,u_2,u_3) ∇ · u → = ∂ u 1 ∂ x + ∂ u 2 ∂ y + ∂ u 3 ∂ z \nabla \cdot \vec{u} = \frac{\partial u_1}{\partial x} + \frac{\partial u_2}{\partial y} + \frac{\partial u_3}{\partial z} ∇ × u → = ( ∂ u 3 ∂ y − ∂ u 2 ∂ z , ∂ u 1 ∂ z − ∂ u 3 ∂ x , ∂ u 2 ∂ x − ∂ u 1 ∂ y ) \nabla \times \vec{u} = \Bigl( \frac{\partial u_3}{\partial y} - \frac{\partial u_2}{\partial z}, ~ \frac{\partial u_1}{\partial z} - \frac{\partial u_3}{\partial x}, ~ \frac{\partial u_2}{\partial x} - \frac{\partial u_1}{\partial y} \Bigr)
Your Turn
Suppose F F F ( x , y ) = a x + b y F(x,y) = ax+by a a b b F ( y ′ , y ) = 0 F(y',y) = 0
Your Turn
Write down an explicit example of a third order, linear, nonconstant coefficient, nonautonomous, nonhomogeneous system of two ODE such that every derivative that could appear, does appear.
Your Turn
Classify the following equations. Are they ODE or PDE? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant coefficient? If it is an ODE, is it autonomous?
∂
2
v
∂
x
2
+
3
∂
2
v
∂
y
2
=
sin
(
x
)
\displaystyle \frac{\partial^2 v}{\partial x^2} + 3 \frac{\partial^2 v}{\partial y^2} = \sin(x)
d
x
d
t
+
cos
(
t
)
x
=
t
2
+
t
+
1
\displaystyle \frac{d x}{dt} + \cos(t) x = t^2+t+1
d
7
F
d
x
7
=
3
F
(
x
)
\displaystyle \frac{d^7 F}{dx^7} = 3F(x)
y
″
+
8
y
′
=
1
\displaystyle y''+8y'=1
x
″
+
t
y
x
′
=
0
,
y
″
+
t
x
y
=
0
\displaystyle x''+tyx'=0, \quad y''+txy = 0
∂
u
∂
t
=
∂
2
u
∂
s
2
+
u
2
\displaystyle \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial s^2} + u^2
Answer
PDE, equation, second order, linear, nonhomogeneous, constant coefficient. ODE, equation, first order, linear, nonhomogeneous, not constant coefficient, not autonomous. ODE, equation, seventh order, linear, homogeneous, constant coefficient, autonomous. ODE, equation, second order, linear, nonhomogeneous, constant coefficient, autonomous. ODE, system, second order, nonlinear. PDE, equation, second order, nonlinear. Your Turn
Write down the general zero th order linear ordinary differential equation. Write down the general solution.
Answer
equation: a ( x ) y = b ( x ) a(x)y=b(x) y = b ( x ) a ( x ) y=\frac{b(x)}{a(x)}
Adapted from Differential Equations for Engineers by Jiří Lebl (https://www.jirka.org/diffyqs/), © Jiří Lebl, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0 .
