How do you use ode45( ) when the equation is not in dy/dt form?

I have an equation that I need to approximate the solution for using ode45( ) with initial values y(0) = 1 and y'(0) = 0 on interval of 0 < t < 150 and then plot x(t)

So far I have this for the equation y" + 4y = sin(1.9t)

syms y(t) 
eqn = diff(y,2) + 4*y == sin(1.9*t)
V = odeToVectorField(eqn) 

I know if the equation was in the form of dy/dt = sin(1.9t) - 4y then it would be

k = @(t,y) sin(1.9*t)-4*y;
[t,y] = oder45(k,[0,150])
plot(t,y)

But I do not know know how to write it for the form that my equation is in at this moment

Answer 1

[Paul]

Check out this link ....

[ssmith]

@Paul Do you have an example that shows how to apply initial conditions?

[Paul]

It's shown on the the doc page at that link. Or check

doc ode45

to see how to specifcy the initial conditions

Answer 2

[Star Strider]

Since the equation as written involves the second derivative, it is also necessary to define the first derivative as a separate element of the resulting system of first-order differential equatiions.

The odeToVectorField function does this (I added the second ‘substitutions’ output to illustrate what the function actually does) —

syms y(t) 
eqn = diff(y,2) + 4*y == sin(1.9*t)

eqn(t) = 

[V,S] = odeToVectorField(eqn) 

V = 

S = 

This is actually

The function presented to ode45 would then be coded using the matlabFunction function.

Explore that process!

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[ssmith]

@Star Strider This is what I have

syms y(t) 
eqn = diff(y,2) + 4*y == sin(1.9*t)
V = odeToVectorField(eqn) 
M = matlabFunction(eqn, 'vars',{'t','Y'})
sol = ode45(M,[0 150])

[Star Strider]

The ode45 call is missing an initial conditions vector. It needs to be either (1x2) or (2x1) for this system (the integration functions don’t care about the orientation). Supply that and the code should work. Since the appear not to have been specified in the original code, they can be anything reasonable that make sense in the context of the equations.

Also, I prefer:

[t,y] = ode45( ... )

since it makes things easier (unless the ‘sol’ structure is required for other reasons).

.

[ssmith]

@Star Strider I am getting error codes with odearguments and ode45. This is my code and I have been following this link as an outline.

syms y(t) 
[V] = odeToVectorField(diff(y,2) + 4*y == sin(1.9*t)) 
M = matlabFunction(V, 'vars',{'t','Y'})
[t,y] = ode45(M,[0 150])

https://www.mathworks.com/help/symbolic/solve-differential-equation-numerically-1.html

[Star Strider]

As I so cogently noted previously ‘The ode45 call is missing an initial conditions vector.’

It still is.

syms y(t) 
[V,S] = odeToVectorField(diff(y,2) + 4*y == sin(1.9*t)) 

V = 

S = 

M = matlabFunction(V, 'vars',{'t','Y'})

M = function_handle with value:
    @(t,Y)[Y(2);sin(t.*(1.9e+1./1.0e+1))-Y(1).*4.0]

[t,y] = ode45(M,[0 150],[1 0]);

figure
plot(t,y)
grid
legend(string(S))

Supplying it solves the problem!

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[ssmith]

@Star Strider What part of the code shows the approximate solution for the initial value on an interval 0 < t < 150 and plot x(t) the graph shows y and Dy

[Star Strider]

The ‘y’ matrix rows correspond to the rows of ‘t’ so the initial values will be the first row ‘t(1)’ (since ‘t’ begins at 0) and the final values are for 150 and would be ‘y(end,:)’ . Unless ‘tspan’ was defined as (0:150) so that the index values corresponded to the ‘t’ values (t(1)=0, y(1,:)=initial values) and (t(151)=150, y(151)=final values) woould there be any direct correspondence between them.

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[ssmith]

@Star Strider is the graph plotted show the phenomenon of beats?

[Star Strider]

Yes, in the sense of ‘beat frequencies’.

Again see my previous Comment for an illustration of the plotted output.

Please read my Comments!

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[ssmith]

@Star Strider ok, thank you!

[Star Strider]

My pleasure!

If my Answer helped you solve your problem, please Accept it!

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