Worksheet: Method of Undetermined Coefficients

In this worksheet, we will practice solving a linear nonhomogeneous differential equation with constant coefficients by using the method of undetermined coefficients.

Q1:

Solve d d     𝑦 π‘₯ + 9 𝑦 = 9 𝑒 .

  • A 𝑦 = 𝐢 3 π‘₯ + 𝐢 3 π‘₯ + 1 2 𝑒     c o s s i n
  • B 𝑦 = 𝐢 3 π‘₯ + 𝐢 3 π‘₯ βˆ’ 1 2 𝑒     c o s s i n
  • C 𝑦 = 𝐢 3 π‘₯ + 𝐢 3 π‘₯ + 𝑒     c o s s i n
  • D 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 1 2 𝑒         
  • E 𝑦 = 𝐢 3 π‘₯ + 𝐢 3 π‘₯ + 9 𝑒     c o s s i n

Q2:

Solve d d d d    𝑦 π‘₯ βˆ’ 1 0 𝑦 π‘₯ + 2 5 𝑦 = 2 5 π‘₯ + 5 π‘₯ + 1 7 .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 2 5 π‘₯ + 5 π‘₯ + 1 7       
  • B 𝑦 = 𝐢 𝑒 + 𝐢 π‘₯ 𝑒 βˆ’ π‘₯ + π‘₯ + 1       
  • C 𝑦 = 𝐢 𝑒 + 𝐢 π‘₯ 𝑒 + π‘₯ + π‘₯ + 1       
  • D 𝑦 = 𝐢 𝑒 + 𝐢 π‘₯ 𝑒 + 2 5 π‘₯ + 5 π‘₯ + 1 7       
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + π‘₯ + π‘₯ + 1       

Q3:

Solve d d d d c o s   𝑦 π‘₯ βˆ’ 2 𝑦 π‘₯ βˆ’ 3 𝑦 = 8 2 π‘₯ .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 3 2 6 5 2 π‘₯ βˆ’ 5 6 6 5 2 π‘₯       s i n c o s
  • B 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 3 2 6 5 2 π‘₯ + 5 6 6 5 2 π‘₯       s i n c o s
  • C 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 3 2 6 5 2 π‘₯ + 5 6 6 5 2 π‘₯       s i n c o s
  • D 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 3 2 6 5 2 π‘₯ + 5 6 6 5 2 π‘₯       c o s s i n
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 8 2 π‘₯       c o s

Q4:

Solve d d c o s   𝑦 π‘₯ βˆ’ 9 𝑦 = 6 3 π‘₯ .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 3 3 π‘₯        c o s
  • B 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 6 3 π‘₯        c o s
  • C 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 1 3 3 π‘₯        c o s
  • D 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 3 3 π‘₯        c o s
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 3 3 π‘₯        s i n

Q5:

Solve d d      𝑦 π‘₯ + 4 𝑦 = 4 𝑒 + 4 π‘₯ .

  • A 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 1 2 𝑒 + π‘₯ + 1 2      c o s s i n
  • B 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 2 𝑒 + π‘₯ βˆ’ 1 2      c o s s i n
  • C 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 1 2 𝑒 + π‘₯ βˆ’ 1 2      c o s s i n
  • D 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 1 2 𝑒 + π‘₯ βˆ’ 1 2          
  • E 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 𝑒 + π‘₯ βˆ’ 1      c o s s i n

Q6:

Solve the third-order differential equation 𝑦 β€² β€² β€² βˆ’ 2 𝑦 β€² β€² + 𝑦 β€² = 2 𝑒 + 2 π‘₯  under the conditions 𝑦 ( 0 ) = 0 , 𝑦 β€² ( 0 ) = 0 , and 𝑦 β€² β€² β€² ( 0 ) = 0 .

  • A 𝑦 = π‘₯ + 4 π‘₯ + 4 + ο€Ή π‘₯ + 4  𝑒   
  • B 𝑦 = π‘₯ + 4 π‘₯ + 4 + ο€Ή π‘₯ + 4  𝑒    
  • C 𝑦 = π‘₯ + 4 π‘₯ + 4 + ο€Ή π‘₯ βˆ’ 4  𝑒   
  • D 𝑦 = π‘₯ + 4 π‘₯ + 4 + ο€Ή π‘₯ βˆ’ 4  𝑒    

Q7:

Find the general solution for the ordinary differential equation 𝑦 β€² β€² βˆ’ 2 𝑦 β€² + 𝑦 = 2 ( π‘₯ ) c o s with constant coefficients whose right-hand side is nonzero.

  • A 𝑦 = ( 𝑐 π‘₯ + 𝑐 ) 𝑒 βˆ’ ( π‘₯ )    s i n
  • B 𝑦 = ( 𝑐 π‘₯ + 𝑐 ) 𝑒 + ( π‘₯ )    s i n
  • C 𝑦 = ( 𝑐 π‘₯ + 𝑐 ) 𝑒 + ( π‘₯ )     s i n
  • D 𝑦 = ( 𝑐 π‘₯ + 𝑐 ) 𝑒 βˆ’ ( π‘₯ )     s i n

Q8:

Solve d d d d   𝑦 π‘₯ βˆ’ 2 𝑦 π‘₯ βˆ’ 3 𝑦 = 2 π‘₯ + 1 .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 2 3 π‘₯ + 1 9      
  • B 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 2 3 π‘₯ βˆ’ 1 9      
  • C 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 2 3 π‘₯ + 1 9      
  • D 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 3 2 π‘₯ βˆ’ 2 3      
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 2 π‘₯ + 1      

Q9:

Solve d d d d      𝑦 π‘₯ + 2 𝑦 π‘₯ + 2 𝑦 = π‘₯ 𝑒 .

  • A 𝑦 = 𝑒 ( 𝐢 π‘₯ + 𝐢 π‘₯ ) + π‘₯ 𝑒 βˆ’ 1 2 𝑒          c o s s i n
  • B 𝑦 = 𝑒 ( 𝐢 π‘₯ + 𝐢 π‘₯ ) + π‘₯ 𝑒 βˆ’ 2 𝑒         c o s s i n
  • C 𝑦 = 𝑒 ( 𝐢 π‘₯ + 𝐢 π‘₯ ) + π‘₯ 𝑒 βˆ’ 2 𝑒          c o s s i n
  • D 𝑦 = 𝑒 ( 𝐢 π‘₯ + 𝐢 π‘₯ ) + 2 π‘₯ 𝑒 βˆ’ 𝑒          c o s s i n
  • E 𝑦 = 𝑒 ( 𝐢 π‘₯ + 𝐢 π‘₯ ) + π‘₯ 𝑒 + 2 𝑒          c o s s i n

Q10:

Solve d d d d     𝑦 π‘₯ βˆ’ 2 𝑦 π‘₯ βˆ’ 3 𝑦 = 𝑒 .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 𝑒        
  • B 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 3 𝑒        
  • C 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 3 𝑒        
  • D 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 3 𝑒        
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 1 3 𝑒        

Q11:

Solve the following differential equation under the conditions 𝑦 ( 0 ) = βˆ’ 1 and 𝑦 β€² ( 0 ) = βˆ’ 2 : 𝑦 β€² β€² + 3 𝑦 β€² + 2 𝑦 = 1 0 ( π‘₯ ) s i n .

  • A 𝑦 = 𝑒 + 𝑒 + ( π‘₯ ) βˆ’ 3 ( π‘₯ )    s i n c o s
  • B 𝑦 = 𝑒 + 𝑒 + ( π‘₯ ) βˆ’ 3 ( π‘₯ )      s i n c o s
  • C 𝑦 = 𝑒 + 𝑒 + ( π‘₯ ) βˆ’ 3 ( π‘₯ )     s i n c o s
  • D 𝑦 = 𝑒 + 𝑒 + ( π‘₯ ) βˆ’ 3 ( π‘₯ )     s i n c o s

Q12:

Solve d d     𝑦 π‘₯ + 4 𝑦 = 4 π‘₯ + 8 π‘₯ + 1 8 π‘₯ + 2 0 .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + π‘₯ + 2 π‘₯ + 3 π‘₯ + 4         
  • B 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + π‘₯ + 2 π‘₯ + 3 π‘₯ + 4     c o s s i n
  • C 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 4 π‘₯ + 3 π‘₯ + 2 π‘₯ + 1     c o s s i n
  • D 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 4 π‘₯ + 8 π‘₯ βˆ’ 6 π‘₯ + 4     c o s s i n
  • E 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 4 π‘₯ + 8 π‘₯ + 1 8 π‘₯ + 2 0     c o s s i n

Q13:

Solve d d d d     𝑦 π‘₯ βˆ’ 3 𝑦 π‘₯ βˆ’ 4 𝑦 = 𝑒 .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 1 5 π‘₯ 𝑒  οŠͺ      
  • B 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 5 π‘₯ 𝑒   οŠͺ     
  • C 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 5 𝑒  οŠͺ      
  • D 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 5 π‘₯ 𝑒  οŠͺ      
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 5 π‘₯ 𝑒  οŠͺ      

Q14:

Solve d d s i n   𝑦 π‘₯ + 4 𝑦 = 8 2 π‘₯ .

  • A 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ βˆ’ 2 π‘₯ 2 π‘₯   c o s s i n c o s
  • B 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 2 π‘₯ 2 π‘₯        c o s
  • C 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 2 π‘₯ 2 π‘₯        c o s
  • D 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 2 π‘₯ 2 π‘₯   c o s s i n s i n
  • E 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 2 π‘₯ 2 π‘₯   c o s s i n c o s

Q15:

Solve d d d d    𝑦 π‘₯ βˆ’ 2 𝑦 π‘₯ + 𝑦 = 6 𝑒 .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 π‘₯ 𝑒 + 3 π‘₯ 𝑒      
  • B 𝑦 = 𝐢 𝑒 + 𝐢 π‘₯ 𝑒 + 6 π‘₯ 𝑒     
  • C 𝑦 = 𝐢 𝑒 + 𝐢 π‘₯ 𝑒 + 3 𝑒     
  • D 𝑦 = 𝐢 𝑒 + 𝐢 π‘₯ 𝑒 + 3 π‘₯ 𝑒     
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 3 𝑒     

Q16:

Find the general solution for the following differential equation using the method of undetermined coefficients: 𝑦 + 3 𝑦 + 2 𝑦 = 𝑒 , 𝑖 = βˆ’ 1 .      

  • A 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 + 1 1 0 ο€Ή 𝑒 βˆ’ 3 𝑖 𝑒            
  • B 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 + 1 1 0 ο€Ή 𝑒 βˆ’ 3 𝑖 𝑒             
  • C 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 + 1 1 0 ο€Ή 𝑒 + 3 𝑖 𝑒            
  • D 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 + 1 1 0 ο€Ή 𝑒 βˆ’ 3 𝑖 𝑒           

Q17:

Solve d d c o s   𝑦 π‘₯ + 𝑦 = π‘₯ π‘₯ .

  • A 𝑦 = 𝐢 π‘₯ + 𝐢 π‘₯ + π‘₯ π‘₯ + π‘₯ π‘₯    c o s s i n c o s s i n
  • B 𝑦 = 𝐢 π‘₯ + 𝐢 π‘₯ + 1 4 ο€Ή π‘₯ π‘₯ + π‘₯ π‘₯     c o s s i n c o s s i n
  • C 𝑦 = 𝐢 π‘₯ + 𝐢 π‘₯ + 1 4 ο€Ή π‘₯ π‘₯ + π‘₯ π‘₯     c o s s i n c o s s i n
  • D 𝑦 = 𝐢 π‘₯ + 𝐢 π‘₯ βˆ’ 1 4 ο€Ή π‘₯ π‘₯ + π‘₯ π‘₯     c o s s i n c o s s i n
  • E 𝑦 = 𝐢 π‘₯ + 𝐢 π‘₯ + 4 ο€Ή π‘₯ π‘₯ + π‘₯ π‘₯     c o s s i n c o s s i n

Q18:

Solve d d s i n   𝑦 π‘₯ + 4 𝑦 = 1 6 βˆ’ 5 3 π‘₯ .

  • A 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 3 π‘₯ + 4   c o s s i n s i n
  • B 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ + 4 3 π‘₯ + 1   c o s s i n s i n
  • C 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ βˆ’ 3 π‘₯ + 4   c o s s i n s i n
  • D 𝑦 = 𝐢 2 π‘₯ + 𝐢 2 π‘₯ βˆ’ 5 1 3 3 π‘₯ + 4   c o s s i n s i n
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 3 π‘₯ + 4        s i n

Q19:

Which of the following is a particular solution to the differential equation 𝑓 ( π‘₯ ) + 𝑓 β€² ( π‘₯ ) + 𝑓 β€² β€² ( π‘₯ ) = 9 π‘₯ + 8 π‘₯ βˆ’ 4 π‘₯ + 2   .

  • A 𝑓 ( π‘₯ ) = 9 π‘₯ βˆ’ 1 9 π‘₯ βˆ’ 2 0 π‘₯ + 6 0  
  • B 𝑓 ( π‘₯ ) = 9 π‘₯ + 3 5 π‘₯ βˆ’ 2 0 π‘₯ βˆ’ 4 8  
  • C 𝑓 ( π‘₯ ) = 9 π‘₯ 4 + 8 π‘₯ 3 βˆ’ 2 π‘₯ + 2 π‘₯ οŠͺ  
  • D 𝑓 ( π‘₯ ) = 2 7 π‘₯ + 1 6 π‘₯ βˆ’ 4 

Q20:

Solve d d d d s i n   𝑦 π‘₯ βˆ’ 2 𝑦 π‘₯ + 5 𝑦 = 1 0 π‘₯ .

  • A 𝑦 = 𝑒 ( 𝐢 2 π‘₯ + 𝐢 2 π‘₯ ) + 1 0 π‘₯    c o s s i n s i n
  • B 𝑦 = 𝑒 ( 𝐢 2 π‘₯ + 𝐢 2 π‘₯ ) βˆ’ 2 π‘₯ + π‘₯    c o s s i n s i n c o s
  • C 𝑦 = 𝑒 ( 𝐢 2 π‘₯ + 𝐢 2 π‘₯ ) + 2 π‘₯ + π‘₯    c o s s i n s i n c o s
  • D 𝑦 = 𝑒 ( 𝐢 2 π‘₯ + 𝐢 2 π‘₯ ) + 2 π‘₯ βˆ’ π‘₯    c o s s i n s i n c o s
  • E 𝑦 = 𝑒 ( 𝐢 2 π‘₯ + 𝐢 2 π‘₯ ) + π‘₯ + 2 π‘₯    c o s s i n s i n c o s

Q21:

Find the general solution for the following nonhomogeneous ordinary differential equation: 𝑦 β€² β€² + 2 𝑦 β€² + 𝑦 = π‘₯ 𝑒    .

  • A 𝑦 = 𝑐 𝑒 + 𝑐 π‘₯ 𝑒 + π‘₯ 𝑒 1 2      οŠͺ  
  • B 𝑦 = 𝑐 𝑒 + 𝑐 π‘₯ 𝑒 + π‘₯ 𝑒 1 2       οŠͺ  
  • C 𝑦 = 𝑐 𝑒 + 𝑐 π‘₯ 𝑒 + π‘₯ 𝑒 1 2       οŠͺ 
  • D 𝑦 = 𝑐 𝑒 + 𝑐 π‘₯ 𝑒 + π‘₯ 𝑒 1 2      οŠͺ  

Q22:

Find the particular solution 𝑦  and the complementary function 𝑦  for the following ordinary differential equation using the method of inverse operators: 𝑦 β€² β€² βˆ’ 𝑦 = π‘₯ + 3 π‘₯ βˆ’ 4  .

  • A 𝑦 = βˆ’ ο€Ή π‘₯ + 9 π‘₯ βˆ’ 4    , 𝑦 = 𝑐 𝑒 + 𝑐 𝑒      
  • B 𝑦 = βˆ’ ο€Ή π‘₯ + 9 π‘₯ βˆ’ 4    , 𝑦 = 𝑐 + 𝑐 𝑒    
  • C 𝑦 = βˆ’ ο€Ή π‘₯ + 9 π‘₯ βˆ’ 4    , 𝑦 = 𝑐 𝑒 + 𝑐 𝑒      
  • D 𝑦 = π‘₯ + 9 π‘₯ βˆ’ 4   , 𝑦 = 𝑐 𝑒 + 𝑐      

Q23:

Solve d d d d    𝑦 π‘₯ βˆ’ 2 𝑦 π‘₯ + 4 𝑦 = π‘₯ .

  • A 𝑦 = 𝑒 ο€» 𝐢 √ 3 π‘₯ + 𝐢 √ 3 π‘₯  + 1 4 π‘₯     c o s s i n
  • B 𝑦 = 𝑒 ο€» 𝐢 √ 3 π‘₯ + 𝐢 √ 3 π‘₯  + π‘₯     c o s s i n
  • C 𝑦 = 𝑒 ο€» 𝐢 √ 3 π‘₯ + 𝐢 √ 3 π‘₯  + 1 4 π‘₯ + 1 4 π‘₯ + 1 4     c o s s i n
  • D 𝑦 = 𝑒 ο€» 𝐢 √ 3 π‘₯ + 𝐢 √ 3 π‘₯  + 1 4 π‘₯ βˆ’ 1 4 π‘₯     c o s s i n
  • E 𝑦 = 𝑒 ο€» 𝐢 √ 3 π‘₯ + 𝐢 √ 3 π‘₯  + 1 4 π‘₯ + 1 4 π‘₯     c o s s i n

Q24:

Solve d d s i n    𝑦 π‘₯ βˆ’ 4 𝑦 = π‘₯ + 3 π‘₯ 𝑒 .

  • A 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 5 π‘₯ + π‘₯ 𝑒 + 2 3 𝑒          s i n
  • B 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 3 π‘₯ βˆ’ π‘₯ 𝑒 βˆ’ 2 3 𝑒          s i n
  • C 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 5 π‘₯ βˆ’ π‘₯ 𝑒 βˆ’ 2 3 𝑒          s i n
  • D 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + π‘₯ + 3 π‘₯ 𝑒         s i n
  • E 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 βˆ’ 1 5 π‘₯ βˆ’ π‘₯ 𝑒 βˆ’ 2 3 𝑒          c o s

Q25:

Solve d d c o s   𝑦 π‘₯ + 9 𝑦 = 5 2 π‘₯ .

  • A 𝑦 = 𝐢 3 π‘₯ + 𝐢 3 π‘₯ + 2 π‘₯   c o s s i n c o s
  • B 𝑦 = 𝐢 3 π‘₯ + 𝐢 3 π‘₯ βˆ’ 2 π‘₯   c o s s i n c o s
  • C 𝑦 = 𝐢 𝑒 + 𝐢 𝑒 + 5 2 π‘₯        c o s
  • D 𝑦 = 𝐢 3 π‘₯ + 𝐢 3 π‘₯ + 5 1 3 2 π‘₯   c o s s i n c o s
  • E 𝑦 = 𝐢 3 π‘₯ + 𝐢 3 π‘₯ + 2 π‘₯   c o s s i n s i n

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.