Using
Matlab Solving Ordinary Differential Equations
Analytically (by
T. Gu)
Click here to see
numerical tools with graphical user interface developed by T. Gu
Matlab can solve linear ODE's or ODE systems with
or without initial/boundary conditions. Do not expect Matlab
to solve nonlinear ODE's which typically have no analytical solutions. Matlab version R12 and later treats y as default function symbol and t as default variable symbol. Earlier
version treats x as default symbol.
Example 1. d2y/dx2 -2dy/dx -3y=x2
>> dsolve('D2y - 2*Dy - 3*y=x^2', 'x')ans =
-14/27+4/9*x-1/3*x^2+C1*exp(3*x)+C2*exp(-x)
Example 2. d2y/dx2 -2dy/dx -3y=x2, with y(0)=0, and y(1)=1.
>> dsolve('D2y - 2*Dy - 3*y=x^2','y(0)=0, y(1)=1','x')ans =
-14/27+4/9*x-1/3*x^2+2/27*(7*exp(-1)-19)/(exp(-1)-exp(3))*exp(3*x)-2/27*(-19+7*exp(3))/(exp(-1)-exp(3))*exp(-x)
Example 3. d2y/dx2 -2dy/dx -3y=x2, with y(0)=0, and dy/dx =1 at x=1
>> dsolve('D2y - 2*Dy - 3*y=x^2','y(0)=0, Dy(1)=1','x')ans =
-1/3*x^2+4/9*x-14/27+1/9*(-11+14*exp(3))/(3*exp(3)+exp(-1))*exp(-x)
+1/27*(33+1 4*exp(-1))/(3*exp(3)+exp(-1))*exp(3*x)
Example 4. d2y/dx2 -2dy/dx -3y=0, with y(0)=a, and y(1)=b.
>> dsolve('D2y-2*Dy-3*y=0','y(0)=a, y(1)=b')ans =
(exp(3)*a-b)/(-exp(-1)+exp(3))*exp(-t)-(-b+a*exp(-1))/(-exp(-1)+exp(3))*exp(3*t)
>> pretty(ans)
(exp(3) a - b) exp(-t) (-b + a exp(-1)) exp(3 t)
---------------------- - -------------------------
-exp(-1) + exp(3) -exp(-1) + exp(3)
Example 5. d2y/dt2+y=4cos(t), y(pi/2)=2pi, dy/dt=-3 at t=pi/2
>> y=dsolve('D2y+y=4*cos(t)' , 'y(pi/2)=2*pi, Dy(pi/2)=-3')y =
-2*sin(t)^2*cos(t)+(2*sin(t)*cos(t)+2*t)*sin(t)+5*cos(t)+pi*sin(t)
>> simplify(y)
ans =
2*sin(t)*t+5*cos(t)+pi*sin(t)
Example 6. Solving two ODE's with two initial conditions:
dx/dt =3x+4y, dy/dt =-4x+3y, with x(0)=0, y(0)=1
>> [x,y]=dsolve('Dx=3*x+4*y','Dy=-4*x+3*y','x(0)=0,y(0)=1')x =
exp(3*t)*sin(4*t)
y =
exp(3*t)*cos(4*t)
Example 7. Bessel Equation: x2(d2y/dx2) +x(dy/dx) + (x2-v2)y=0
Solution: y=c1Jv(x) + c2Yv(x)
>> y=dsolve('x^2*D2y+x*Dy+(x^2 - v^2)*y =0','x')y =
C1*besselj(v,x)+C2*bessely(v,x)
Using Matlab Solving Algebraic Equations Analytically
Matlab can be used to solve
nonlinear algebraic equations or equation systems. Sometimes only numeric
answers are returned. Sometimes, Matlab fails.
Example 1:
>>syms x
>>solve(-x^3 +3*x ==0)
ans =
[ 0]
[3^(1/2)]
[-3^(1/2)]
>> numeric(ans)
ans =
0
-1.7321
1.7321
Or, use the command "roots" for polynomials
>> roots([-1,0,3,0])
ans =
0
1.7321
-1.7321
Example 2:
>>syms x a b
>> solve('log(x)+a^2==b')ans =
exp(-a^2+b)
Example 3:
>> T=solve('ln(T)+a^2=b','T')T =
exp(-a^2+b)
Example 4: Solving 2 equations simultaneously
>> syms x1 x2
>> [x1,x2]=solve(x1^2+x2==10,x1-x2==2)
x1 =
3
-4
x2 =
1
-6
Using MATLAB to Solve for Friction Factor Using Colebrook Formula
Save the following 6 lines into friction.m m-file.
function dummy %this allows you to pack the 5 lines below into a single m-file.
fsolve(@cole, 0.02) %fsolve looks for cole function below. If missing, it looks in current directory.
function y=cole(f)
eoverD=0.000375;
Re=1.37e+4;
y=1/sqrt(f)+2*log10(eoverD/3.7 + 2.51/Re/sqrt(f));
Run friction.m in Matlab
>>friction
f =
0.0291
Using fsolve for a nonlinear equation system
The problem below is from a multiple pipe system in ChE345 (Fluid Mechanics.)
Save the 7 lines below as a file named fluids.m to be run in Matlab.
function dummy %This allows you to pack the 6 lines below into a single m-file
x0=[1,1,1];
fsolve(@mflow,x0) %fsolve looks for mflow function below. If missing, it looks in current directory.
function z=mflow(v)
z(1)=v(1)-v(2)-v(3);
z(2)=322 -v(1)^2-0.4*v(3)^2;
z(3)=258 -v(1)^2-0.5*v(2)^2;
Run the m-file to get three velocities.
>> fluids
ans =
15.9327 2.8802 13.0526
The same problem can be solved using the solve command.
Save the 5 lines in equations.m m-file.
syms x1 x2 x3
e1=x1 == x2 + x3;
e2=322 == x1^2+0.4*x3^2;
e3=258 == x1^2+0.5*x2^2;
[x1,x2,x3]=solve([e1,e2,e3]);
>>equations
>>eval([x1,x2,x3])
ans =
15.9327 2.8802 13.0526
5.6656 -21.2556 26.9212
-15.9327 -2.8802 -13.0526
-5.6656 21.2556 -26.9212
"solve" is not as reliable as "fsolve" in MATLAB.
When you copy text from WORD or html to Matlab, be careful! You may carry over some hidden symbols. Use a .txt file.
Another example using fsolve to solve 6 nonlinear equations
This is for Problem 8.64 on p. 374 of ChE345 (Fluid Mechanics) textbook.
How to solve Homework Problem 8.64* using Matlab's fsolve?
Save the 10 text lines below as solution.m file.
function dummy
x0=[1,1,1,0.02,0.02,0.02];
fsolve(@fcn,x0)
function z=fcn(x)
z(1)=x(1)-0.64*x(2)-0.64*x(3);%x(1)=v1, x(2)=v2, x(3)=v3
z(2)=40-102*x(4)*x(1)^2 - 127*x(5)*x(2)^2; %x(4)=f1, x(5)=f2, x(6)=f3
z(3)=60-102*x(4)*x(1)^2 - 255*x(6)*x(3)^2;
z(4)=1/sqrt(x(4))+2.0*log10(1.22e-4 +2.81e-5/x(1)/sqrt(x(4))); %Colebrook for f1
z(5)=1/sqrt(x(5))+2.0*log10(1.52e-4 +3.52e-5/x(2)/sqrt(x(5))); %Colebrook for f2
z(6)=1/sqrt(x(6))+2.0*log10(1.52e-4 +3.52e-5/x(3)/sqrt(x(6))); %Colebrook for f3
Run solution.m in Matlab
>>solution
ans =
3.5001 2.6915 2.7774 0.0179 0.0192 0.0191
The answers are:
v1=3.500 m/s, v2=2.692 m/s, v3=2.777 m/s
f1=0.0179, f2=0.0192, f3=0.0191
Factoring a function or
number
>> factor(12)
ans =
2 2 3
>> syms x y % declare
that x and y are symbols
>> factor(x^2-y^2)
ans =
(x-y)*(x+y)
>> factor(x^2-1)
ans =
(x-1)*(x+1)