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Hello Maple enthusiasts,

I am excited to share a sample worksheet on Ordinary Differential Equations (ODEs), created as part of my ongoing project—a book I am writing for undergraduate students. This book is designed to teach ODEs using Maple, offering an interactive and intuitive approach to solving differential equations.

As far as I know, there aren’t many books available in the Greek language that combine ODEs with Maple. In fact, I believe there’s only one other such resource, which highlights the lack of materials in this niche. My goal is to fill this gap by providing students and educators with a resource that is both practical and accessible, leveraging Maple's powerful capabilities to deepen understanding and simplify complex concepts.

The worksheet I’m sharing includes:

  • Step-by-step solutions to ODEs using Maple.
  • Graphical representations to visualize solutions, which I believe are invaluable for fostering comprehension.

I hope this preview sparks your interest and provides insight into the teaching style and structure of the upcoming book. I would love to hear your thoughts, feedback, or suggestions for topics you think should be included.

Solve the following differential equation

diff(y(x), x) = x*y(x)
with initial condition y(0) = 1

> 

restart; with(plots); with(DEtools)
> 

ode := diff(y(x), x) = x*y(x)

diff(y(x), x) = x*y(x)

 (1)
> 

ic := y(0) = 1

y(0) = 1

 (2)
> 

general_solution := dsolve(ode, y(x))

y(x) = c__1*exp((1/2)*x^2)

 (3)
> 

particular_solution := dsolve({ic, ode}, y(x))

y(x) = exp((1/2)*x^2)

 (4)
> 

soln := dsolve({ic, ode}, y(x), numeric)

``

 (5)
> 

p1 := odeplot(soln, [x, y(x)], x = -5 .. 5, labels = ["x", "y(x)"], color = green)
> 

directionfield := dfieldplot(ode, [y(x)], x = -5 .. 5, y = -5 .. 5, color = blue, arrows = slim, scaling = constrained, axes = boxed)
> 

display([p1, directionfield], view = [-5 .. 5, -5 .. 5])

 

NULL

Solve the following differential equation

diff(y(x), x) = x/y(x)
with initial condition y(0) = 2

> 

restart; with(plots); with(DEtools)
> 

ode1 := diff(y(x), x) = x/y(x)

diff(y(x), x) = x/y(x)

 (6)
> 

ic := y(0) = 2

y(0) = 2

 (7)
> 

dsolve(ode1, y(x))

y(x) = (x^2+c__1)^(1/2), y(x) = -(x^2+c__1)^(1/2)

 (8)
> 

NULL
> 

soln := dsolve({ic, ode1}, y(x), numeric)
> 

p1 := odeplot(soln, [x, y(x)], x = -5 .. 5, labels = ["x", "y(x)"], color = green)
> 

directionfield := dfieldplot(ode1, [y(x)], x = -5 .. 5, y = -5 .. 5, color = blue, arrows = slim, scaling = constrained, axes = boxed)
> 

display([p1, directionfield], view = [-5 .. 5, -5 .. 5])

 

NULL

Solve the following differential equation

dy/dx = e^y/(x^2+1)
with initial condition y(0) = -1

> 

restart; with(plots); with(DEtools)
> 

ode2 := diff(y(x), x) = exp(y(x))/(x^2+1)

diff(y(x), x) = exp(y(x))/(x^2+1)

 (9)
> 

ic := y(0) = -1

y(0) = -1

 (10)
> 

dsolve(ode2, y(x))

y(x) = ln(-1/(arctan(x)+c__1))

 (11)
> 

soln := dsolve({ic, ode2}, y(x), numeric)
> 

p1 := odeplot(soln, [x, y(x)], x = -5 .. 5, labels = ["x", "y(x)"], color = green)
> 

directionfield := dfieldplot(ode2, [y(x)], x = -5 .. 5, y = -5 .. 5, color = blue, arrows = slim, scaling = constrained, axes = boxed)
> 

display([p1, directionfield], view = [-5 .. 5, -5 .. 5])

 

NULL

Solve the following differential equation

dy/dx = y^2+y
with initial condition y(1) = 2

> 

restart; with(plots); with(DEtools)
> 

ode3 := diff(y(x), x) = y(x)+y(x)^2

diff(y(x), x) = y(x)+y(x)^2

 (12)
> 

ic := y(1) = 2

y(1) = 2

 (13)
> 

dsolve(ode3, y(x))

y(x) = 1/(-1+exp(-x)*c__1)

 (14)
> 

soln := dsolve({ic, ode3}, y(x))

y(x) = 2/(-2+3*exp(-x)*exp(1))

 (15)
> 

DEplot(ode3, y(x), x = -5 .. 5, y = -5 .. 5, [[y(1) = 2]], color = blue, arrows = slim, scaling = constrained, axes = boxed)

 

NULL

Solve the following differential equation

y*dy/dx-x = 0
with initial condition y(0) = 4, y(1) = 2, y(-1) = -2and y(-2) = -4.

> 

restart; with(plots); with(DEtools)
> 

ode4 := y(x)*(diff(y(x), x))-x = 0

y(x)*(diff(y(x), x))-x = 0

 (16)
> 

ic := y(0) = 4

y(0) = 4

 (17)
> 

dsolve(ode4, y(x))

y(x) = (x^2+c__1)^(1/2), y(x) = -(x^2+c__1)^(1/2)

 (18)
> 

DEplot(ode4, y(x), x = -5 .. 5, y = -5 .. 5, [[y(1) = 2], [y(0) = 4], [y(-1) = -2], [y(-2) = -4]], color = blue, arrows = slim, scaling = constrained, axes = boxed)

 

 

 

NULL

Solve the following differential equation

dy/dx = 2*x+2*xy^2/y
with initial condition y(0) = 2.

> 

restart; with(plots); with(DEtools)
> 

ode5 := diff(y(x), x) = (2*x+2*x*y(x)^2)/y(x)

diff(y(x), x) = (2*x+2*x*y(x)^2)/y(x)

 (19)
> 

ic := y(0) = 2

y(0) = 2

 (20)
> 

dsolve(ode5, y(x))

y(x) = (exp(2*x^2)*c__1-1)^(1/2), y(x) = -(exp(2*x^2)*c__1-1)^(1/2)

 (21)
> 

psoln := dsolve({ic, ode5}, y(x))

y(x) = (5*exp(2*x^2)-1)^(1/2)

 (22)
> 

soln := dsolve({ic, ode5}, y(x), numeric)
> 

p1 := odeplot(soln, [x, y(x)], x = -5 .. 5, labels = ["x", "y(x)"], color = green)
> 

directionfield := dfieldplot(ode5, [y(x)], x = -5 .. 5, y = -3 .. 10, color = blue, arrows = slim, scaling = constrained, axes = boxed)
> 

display([p1, directionfield], view = [-5 .. 5, -3 .. 10])

 
> 

NULL


 

Download separable_diff_equations.mw

 
ode differential-equations initial-conditions plotting

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