Differential Equations Compute y’ and y’’ and then combine these derivatives with y as a linear...

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Differential Equations Compute y’ and y’’ and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y’, y’’)=0. The symbols c1 and c2 represent constants. Y= c1ex + c2xex Verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.
	
Document Preview: Differential EquationsCompute y’ and y’’ and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y’, y’’)=0. The symbols c1 and c2 represent constants. 							Y= c1ex + c2xex Verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.y’’ + y = sexx;  y=xsinx + (cosx)ln(cosx)Solve the differential equation (2x+y+1)y’ =1Solve the given initial-value problem and give the largest interval I on which the solution is defined. 			dy/dt + 2(t+1)y2 = 0, y(0) = -1/8According to Stefan’s law of radiation the absolute temperature T of a body cooling in a medium at constant absolute temperature Tm is given bydT/dt = k(T4 – T4m),  solve the differential equation.Using the information provided in problem #5, show that when T-Tm is small in comparison to Tm then Newton’s law of cooling approximates Stefan’s law.Find the general solution of the differential equation.  2y’’ + 2y’ +3y =0Solve the differential equation subject to the indicated conditions.Y’’ + 2y’ +y=0, y(-1) = 0, y’(0) = 0Amass weighing 32 pounds stretches a spring 6 inches. The mass moves thought a medium offering a damping force that is numerically equal to ? times the instantaneous velocity. Determine the values of ? >0 for which the spring? Mass system will exhibit oscillatory motion.The vertical motion of a mass attached to a spring is describe by the IVP 1/4x’’ + x’ + x = 0, x(0)=4, x’(0)=2. Determine the maximum vertical displacement of the mass.

Robert · Accepted Answer

Sol: (1) 
1 2The symbol  and   reptresent constants,C C  
                          1 2                                  ..................... (1),
x xY C e C xe   
              On taking a differentiation with respect to x ,                              
                       
 1 2
1 2 2                               ........................ (2),
x x x
x x x
Y C e C xe e
Y C e xe C C e
   
   
 
           From equation (1) and (2), we get                    
                          2                                              ......................... (3),
xY Y C e    
   On taking a differentiation with respect to x ,                                    
                         2                                         .........................(4),
xY Y C e    
 From equation (3),   2  =
xC e Y Y    
 From equation (4),
2  
2 0 
Y Y Y Y
Y Y Y
Y Y Y
    
  
   

Sol: (2) The given differential equation is 
                                   secY Y x    
The particular solution is
   
   
   
2
1
. sec
1
1
sec
1 1 1
sec
2
1 1 1 1
. sec sec                   ...............  (1),
2 2
P I x
D
x
D i D i
x
i D i D i
P I x x
i D i i D i



 
 
  
  
 
 
 
Now,
 
 
 
1
sec sec
cos sin
cos
1 tan
ln cos
ix ix
ix
ix
ix
x e e xdx
D i
x i x
e dx
x
e i x dx
e x i x




 
 



 
Changing  to ,  we havei i  
                     
 
 
1
sec ln cosixx e x i x
D i
 

 
Putting these values in equation (1), we get
   
1 1
. ln cos ln cos
2 2
ln cos ln cos
2 2 2 2
ln cos
2 2
. sin cos ln cos
ix ix
ix ix
ix ix
ix ix ix ix
P I e x i x e x i x
i i
x e x x e x
e e
i i
e e e e
x x
i
P I x x x x



 
         
   
    
    
   
 
 
Verified. 
                 Interval: , 
Sol: (3)   The given differential equation is 
                                   2 1 1x y y                                     …………….. (1), 
Let
   
   
   
 2
2
2
v x y x x
dv x dy x
dx dx
dy x dv x
dx dx
 
 
 
 
From equation (1), we get 
                              
 
 1 2 1
dv x
v x
dx
 
   
 
 
Solve for
 dv x
dx
:

Differential Equations Compute y’ and y’’ and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y’,...

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