Differential Equations

4224 days ago by comphy

load "http://dl.dropbox.com/u/1092631/init.sage" 
        
Custom Commands:
  • R: Round Big
  • r: Round Small
  • rsm: Round Very Small
  • raddeg(radians): Radians to Degrees
  • degrad(degrees): Degrees to Radians
  • table(function, low, high, step): Prints an html table showing values
        of a function from low to high at each step
  • calc(function, [values]): Prints values of a function at a set of points
  • nplot(function): Plots a function from -10 to 10
  • bplot(function): Plots a function from (-10,-10) to (10,10)
  • Variables: x,y,z,t

                                
                            

                            
                                

                                    
Custom Commands:
  • R: Round Big
  • r: Round Small
  • rsm: Round Very Small
  • raddeg(radians): Radians to Degrees
  • degrad(degrees): Degrees to Radians
  • table(function, low, high, step): Prints an html table showing values
        of a function from low to high at each step
  • calc(function, [values]): Prints values of a function at a set of points
  • nplot(function): Plots a function from -10 to 10
  • bplot(function): Plots a function from (-10,-10) to (10,10)
  • Variables: x,y,z,t

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
y1=e^(3*x)
y2=e^(-1*x)
f=2*sin(x)*sin(x)
show((y2 * f))
show(wronskian(y1,y2))
show((y2 * f)/wronskian(y1,y2))
show(-integral((y2*f)/wronskian(y1,y2),x)) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, e^{\left(-x\right)} \sin\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}-4 \, e^{\left(2 \, x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, e^{\left(-3 \, x\right)} \sin\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{156} \, {\left(6 \, \sin\left(2 \, x\right) - 9 \, \cos\left(2 \, x\right) + 13\right)} e^{\left(-3 \, x\right)}

                                
                            

                            
                                

                                    
\newcommand{\Bold}[1]{\mathbf{#1}}2 \, e^{\left(-x\right)} \sin\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}-4 \, e^{\left(2 \, x\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, e^{\left(-3 \, x\right)} \sin\left(x\right)^{2}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{156} \, {\left(6 \, \sin\left(2 \, x\right) - 9 \, \cos\left(2 \, x\right) + 13\right)} e^{\left(-3 \, x\right)}

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
x = var('x')
y= function('y',x)
desolve(diff(y,x,2)+4*diff(y,x)==34*cos(x),y) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
k2*e^(-4*x) + k1 + 8*sin(x) - 2*cos(x)

                                
                            

                            
                                

                                    
k2*e^(-4*x) + k1 + 8*sin(x) - 2*cos(x)

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
solve(2-4*x==(x+37)/4,x) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
[x == (-29/17)]

                                
                            

                            
                                

                                    
[x == (-29/17)]

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
A = matrix([[3,3],[1,5]])
show(A.eigenvectors_right()) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(6, \left[\left(1,\,1\right)\right], 1\right), \left(2, \left[\left(1,\,-\frac{1}{3}\right)\right], 1\right)\right]

                                
                            

                            
                                

                                    
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(6, \left[\left(1,\,1\right)\right], 1\right), \left(2, \left[\left(1,\,-\frac{1}{3}\right)\right], 1\right)\right]

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
O = matrix([[e^(6*t),e^(2*t)],[e^(6*t),-1/3*e^(2*t)]])
Oi = O.inverse()
G = vector([8,4*e^(3*t)])
show(G)
show(Oi) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
\newcommand{\Bold}[1]{\mathbf{#1}}\left(8,\,4 \, e^{\left(3 \, t\right)}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr}
\frac{1}{4} \, e^{\left(-6 \, t\right)} & \frac{3}{4} \, e^{\left(-6 \, t\right)} \\
\frac{3}{4} \, e^{\left(-2 \, t\right)} & -\frac{3}{4} \, e^{\left(-2 \, t\right)}
\end{array}\right)

                                
                            

                            
                                

                                    
\newcommand{\Bold}[1]{\mathbf{#1}}\left(8,\,4 \, e^{\left(3 \, t\right)}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr}
\frac{1}{4} \, e^{\left(-6 \, t\right)} & \frac{3}{4} \, e^{\left(-6 \, t\right)} \\
\frac{3}{4} \, e^{\left(-2 \, t\right)} & -\frac{3}{4} \, e^{\left(-2 \, t\right)}
\end{array}\right)

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
show(Oi*G) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
\newcommand{\Bold}[1]{\mathbf{#1}}\left(2 \, e^{\left(-6 \, t\right)} + 3 \, e^{\left(-3 \, t\right)},\,6 \, e^{\left(-2 \, t\right)} - 3 \, e^{t}\right)

                                
                            

                            
                                

                                    
\newcommand{\Bold}[1]{\mathbf{#1}}\left(2 \, e^{\left(-6 \, t\right)} + 3 \, e^{\left(-3 \, t\right)},\,6 \, e^{\left(-2 \, t\right)} - 3 \, e^{t}\right)

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
U1 = integral(Oi[0]*G,t)
U2 = integral(Oi[1]*G,t)
U = vector([U1,U2])
show(U1)
show(U2)
show(U)
show(O*U) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{3} \, e^{\left(-6 \, t\right)} - e^{\left(-3 \, t\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}-3 \, e^{\left(-2 \, t\right)} - 3 \, e^{t}
\newcommand{\Bold}[1]{\mathbf{#1}}\left(-\frac{1}{3} \, e^{\left(-6 \, t\right)} - e^{\left(-3 \, t\right)},\,-3 \, e^{\left(-2 \, t\right)} - 3 \, e^{t}\right)

                                
                            

                            
                                

                                    
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{3} \, e^{\left(-6 \, t\right)} - e^{\left(-3 \, t\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}-3 \, e^{\left(-2 \, t\right)} - 3 \, e^{t}
\newcommand{\Bold}[1]{\mathbf{#1}}\left(-\frac{1}{3} \, e^{\left(-6 \, t\right)} - e^{\left(-3 \, t\right)},\,-3 \, e^{\left(-2 \, t\right)} - 3 \, e^{t}\right)

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
n = var('n')
integral(x*sin(n*pi*x),x) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
-(pi*n*x*cos(pi*n*x) - sin(pi*n*x))/(pi^2*n^2)

                                
                            

                            
                                

                                    
-(pi*n*x*cos(pi*n*x) - sin(pi*n*x))/(pi^2*n^2)

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
integral(cos(x),x) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
sin(x)

                                
                            

                            
                                

                                    
sin(x)

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
k,n,x = var('k,n,x')
f=(10*sin(3*n)/(n+pi))*e^(i*n*x)
show(f) 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{10 \, e^{\left(i \, n x\right)} \sin\left(3 \, n\right)}{\pi + n}

                                
                            

                            
                                

                                    
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{10 \, e^{\left(i \, n x\right)} \sin\left(3 \, n\right)}{\pi + n}