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We can plot the function as shown in Figure 6-9 EXAMPLE 6-10 Find the general solution of: dy y = dt 1 t2 and plot over 1 < t < 1 for the constant C1 = 0, 10, 20, 30 on the same graph
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MATLAB Demysti ed
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A plot of the solution to
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SOLUTION 6-10 We readily obtain the solution using dsolve:
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>> s = dsolve('Dy = y/sqrt(1 t^2)') s = C1*exp( asin(t))
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We can generate multiple curves for different values of C1 on the same graph using a for loop We create a loop index i and have it assume the values 0, 10, 20, 30 Then we use the subs command to substitute the current value of i and plot the result The for loop looks like this:
>> for i=0:10:30 f = subs(s,'C1',i); ezplot(f,[ 1,1]) hold on end
CHAPTER 6 Symbolic Calculus Differential Eqs
The first line sets our loop variable to i = 0, 10, 20, 30 using the syntax i=start: increment:finish Next we use subs to tell MATLAB to replace C1 by the current value of the loop variable:
f = subs(s,'C1',i);
Then we use ezplot to put the curve on the graph By adding the statement hold on, we tell MATLAB to plot to the same figure each time through the loop So we end up plotting the functions
0, 10*exp( asin(t)), 20*exp( asin(t)), 30*exp( asin(t))
The end statement marks the end of the for loop Next we call hold off so that we close plotting to the current figure and give the plot a title:
>> hold off >> title('IVP Solutions')
The result is shown in Figure 6-10
IVP Solutions 120
0 1
08
06
04
02
Figure 6-10
Plot generated using for loop to place multiple curves on a single figure
EXAMPLE 6-11 Find a solution of dy = 2 t y 2 dt
MATLAB Demysti ed
Plot the solution for different initial values, letting y(0) = 02,,20 in increments of 02 SOLUTION 6-11 First we solve the equation:
>> f = dsolve('Dy= 2*t*y^2','y(0)=y0') f = 1/(t^2+1/y0)
We have told MATLAB to set the initial value to a symbol we denoted y0 Now we can write a for loop to substitute the values y(0) = 02, ,20 First we define our for loop and loop variable, specifying the start, increment, and end point:
for i = 02:02:2
Now we tell MATLAB to substitute i for y0 in the solution:
temp = subs(f,'y0',i);
Next we plot it:
ezplot(temp)
Finally, we close out the loop by telling MATLAB to hold on so we can add a curve to the same figure each time through the loop, then we end it:
hold on end
After the loop runs, we can use the axis command to set the range over each axis to a desired value:
>> axis([ 4 4 0 25]) >> hold off
Don t forget to call hold off so that MATLAB will stop sending data to the same figure The result of all this is show in Figure 6-11
CHAPTER 6 Symbolic Calculus Differential Eqs
1/(t2 + 1/2)
0 4
Figure 6-11
= 2 t y with initial conditions given by A plot of the solution to dt y(0) = 02, , 20 in increments of 02
Systems of Equations and Phase Plane Plots
In this section we will consider the equation for a mass spring system and see how to generate a phase plane plot for the solution First, how can we use MATLAB to generate a solution to a system of differential equations The answer is we pass each equation to dsolve Consider the simple system:
dX dY = Y, = X dt dt X ( 0 ) = 1, Y ( 0 ) = 2
The command to enter the system and solve it is:
>> s = dsolve('DX = Y','DY = X','X(0)= 1','Y(0)=2');
The solution is returned as a vector We can extract the solutions by writing:
>> sX
MATLAB Demysti ed
ans = cos(t)+2*sin(t)
>> sY ans = sin(t)+2*cos(t)
We can show both solutions on one plot:
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