<Text-field style="Heading 1" layout="Heading 1">Calculating Differential Equations</Text-field>

Differential Equations

Name:

The following command must be used to bring in Differential Equations tools

restart:with(DEtools):

You can store a differential equation with the following command.

eqn1:=diff(y(x),x)=y(x);

The following command will attempt to find an analytical solution to the differential equation

dsolve(eqn1);

More Examples

eqn2:=diff(y(x),x)=cos(x);

dsolve(eqn2);

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dsolve(eqn3);

eqn4:=diff(y(x),x)=1;

dsolve(eqn4);

In order to find a unique solution with an initial condition, the IC can be entered into the dsolve command as follows.

dsolve({eqn4,y(0)=1});

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<Text-field style="Heading 1" layout="Heading 1">Finding Slope Fields</Text-field>

restart:with(DEtools):

You can store a differential equation with the following command.

eqn1:=diff(y(x),x)=y(x);

The following command will plot the slope filed of a differential equation

DEplot(eqn1,y(x),x=-10..10,y=-10..10);

The following command will plot a solution curve along with the slope field for a given initial condition

DEplot(eqn1,y(x),x=-10..10,y=-10..10,{[0,1]},linecolor=black);

The following commands are solutions to the handout from class.

eqn2:=diff(y(x),x)=cos(x);

DEplot(eqn2,y(x),x=-10..10,y=-10..10);

DEplot(eqn3,y(x),x=-10..10,y=-10..10);

eqn4:=diff(y(x),x)=1;

DEplot(eqn4,y(x),x=-10..10,y=-10..10);

eqn5:=diff(y(x),x)=x/y(x);

DEplot(eqn5,y(x),x=-10..10,y=-10..10);

eqn6:=diff(y(x),x)=x-y(x);

DEplot(eqn6,y(x),x=-10..10,y=-10..10);

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<Text-field style="Heading 1" layout="Heading 1">Euler's Method</Text-field>

restart:with(DEtools):

The following commands will perform 10 steps of Euler's method.

x_i:=0;y_i:=0;b:=1;N:=10;

X[0]:=x_i;Y[0]:=y_i;h:=(b-x_i)/N;

for n from 0 to N-1 do X[n+1]:=X[n]+h: Y[n+1]:=Y[n]+h*(X[n]+Y[n]): print(n,X[n+1],Y[n+1]); end do:

print(N,X[N],evalf(Y[N]));

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The following shows how a spreadsheet can be used to calculate Euler's Method for the differential equation 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

with an initial condition of (1,0). At the end of the spreadsheet the approximate solution is compared to the actual solution. To see what equations were entered into each cell simply double click on the cell.

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Let's verify what we got from the spreadsheet by modifying our code form above.

restart:

x_i:=1;y_i:=0;b:=2;N:=20;

X[0]:=x_i;Y[0]:=y_i;h:=(b-x_i)/N;

for n from 0 to N-1 do X[n+1]:=X[n]+h: Y[n+1]:=Y[n]+h*(1/X[n]): pring(n,X[n+1],Y[n+1]); end do:

print(N,X[N],evalf(Y[N]));

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<Text-field style="Heading 1" layout="Heading 1">Exercises</Text-field>

For the following exercises we will be using the differential equation 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 and the initial condition (1,1).

<Text-field style="Heading 2" layout="Heading 2">Find the slope field and solution curve for the above differntial equation.</Text-field>

restart:with(DEtools):

JSFH

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<Text-field style="Heading 2" layout="Heading 2">Find Euler's approximation with 20 steps to compute <Font italic="true">y</Font>(2) using a spreadsheet. Verify your answer by modifying the code in the example.</Text-field>

restart:

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JSFH

<Text-field style="Heading 2" layout="Heading 2">Find the exact solution to the differential equation and use it to find the exact value for y(2). Compare that value to what you got using Euler's Method. Use more steps in Euler's Method to gain a better approximation. You can use any way of calculating Euler's method in Maple.</Text-field>

restart:

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