Use the Laplace transform to solve the following differential equations:1. y'+6y=2*t+3, y(0)=12....

Question

Use the Laplace transform to solve the following differential equations:1. y'+6y=2*t+3, y(0)=12. y'-2*y=exp(-t)*cos(t) , y(0)=-23. y''-y=2*t , y(0)=0, y'(0)=-1

David · Accepted Answer

Ans. 1 
 
The given differential equation, 
( ) 6 2 3
d
y y t
dt
    
The above equation is of first order differential equation, 
Its standard form is as, 
' ( ) ( ) ( )y x p x y q x   
The general solution of the above equation is, 
( )
( )
( )
( )
p x dx
p x dx
e q x dx C
y x
e
 



  
In the given equation compare it to the standard form, 
( ) 6, ( ) 2 3p x q x t    
The integrating factor, 
( )
6
p x t
t
I e
I e


 
Now, 
6 ( ) 2 3
d
y y t
dt
    
Multiply 6te in the above equation on both side, 
6 6 6 6.6 ( ) .2 .3t t t t
d
e y e y e t e
dt
    
By the integration by parts, 
6 6 6( ) .2 .3t t t
d
e y e t e
dt
   
Now,  
Solve the L.H.S. 
6 6 6
6 6 6
6 6 6 6
1
( ) .2 .

Use the Laplace transform to solve the following differential equations: 1. y'+6y=2t+3, y(0)=1 2. y'-2y=exp(-t)cos(t) , y(0)=-2 3. y''-y=2t , y(0)=0, y'(0)=-1

Solution

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Use the Laplace transform to solve the following differential equations: 1. y'+6y=2*t+3, y(0)=1 2. y'-2*y=exp(-t)*cos(t) , y(0)=-2 3. y''-y=2*t , y(0)=0, y'(0)=-1

Solution

Answer To This Question Is Available To Download

Related Questions & Answers

Submit New Assignment

Use the Laplace transform to solve the following differential equations: 1. y'+6y=2t+3, y(0)=1 2. y'-2y=exp(-t)cos(t) , y(0)=-2 3. y''-y=2t , y(0)=0, y'(0)=-1