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Differential Equations
  with MATLAB

• Preface
• Contents
• Samples
• Ordering
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Differential Equations
  with Mathematica

• Preface
• Contents
• Samples
• Ordering

Differential Equations
  with Maple

• Preface
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Differential Equations with MATLAB

Series Solutions in MATLAB 2020a and later

As of MATLAB 2020a, the ability to request series solutions to differential equations using dsolve now exists, but the syntax is slightly different from what we guessed it would be when the 2019 edition of Differential Equations with MATLAB was written. On this page, we explain the correct syntax and give some actual examples.

To request a series solution to a differential equation using dsolve, begin with the ordinary dsolve code, but add 'ExpansionPoint' followed by the point around which one wants a series solution. Usually this will be the point at which the initial condition is specified. Specify 'Order' to change the number of terms in the series, just as you would with the series command. We give a number of examples.

♦ Solving y' = y with initial condition y(0)=1. The solution is the exponential function.

 

syms y(t); dsolve(diff(y)==y, y(0)==1, 'ExpansionPoint', 0)

 

This produces the output

 

ans =

 

t^5/120 + t^4/24 + t^3/6 + t^2/2 + t + 1

♦ Finding the series expansion of the Bessel function J₀ by solving Bessel's equation with initial conditions y(0)=1, y'(0)=0. Note that since the solution is even, we need a higher order to get a reasonable number of terms.

 

dsolve(t^2*diff(y,2) + t*diff(y) + t^2*y == 0, y(0)==1, ...

subs(diff(y),t,0)==0, 'ExpansionPoint', 0, 'Order', 8)

 

This produces the output

 

ans =

 

- t^6/2304 + t^4/64 - t^2/4 + 1

 

♦ Solving 2ty'' + y' + ty = 0 in Frobenius series around t = 0.

 

dsolve(2t*diff(y,2) + diff(y) + t*y == 0, ...

'ExpansionPoint', 0)

 

This produces the output

 

ans =

 

               t^4/168 - t^2/6 + 1

t^(1/2) - t^(5/2)/10 + t^(9/2)/360