Differential Equations MTH 242 Lecture 11 Dr Manshoor

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Differential Equations MTH 242 Lecture # 11 Dr. Manshoor Ahmed

Differential Equations MTH 242 Lecture # 11 Dr. Manshoor Ahmed

Summary (Recall) • Construction of 2 nd solution from a known solution. • Reduction

Summary (Recall) • Construction of 2 nd solution from a known solution. • Reduction of order. • How to solve a homogeneous linear differential with constant coefficients? • Auxiliary or characteristics equation. • How to write the general of solution of linear differential with constant coefficients

Solution of non-homogeneous DE •

Solution of non-homogeneous DE •

Solution of non-homogeneous DE •

Solution of non-homogeneous DE •

The Method of Undetermined Coefficients •

The Method of Undetermined Coefficients •

Caution! • In addition to the form of the input function , the educated

Caution! • In addition to the form of the input function , the educated guess for must take into consideration the functions that make up the complementary function. • No function in the assumed must be a solution of the associated homogeneous differential equation. This means that the assumed should not contain terms that duplicate terms in.

Trial particular solutions No 1. 2. 3. 4. Terms Choice for

Trial particular solutions No 1. 2. 3. 4. Terms Choice for

Some functions and their particular solutions 1. 2. 1(Any constant) 3. 4. 5. 6.

Some functions and their particular solutions 1. 2. 1(Any constant) 3. 4. 5. 6. 7. 8. Form of

9. 10. 11. 12.

9. 10. 11. 12.

If equals a sum? Suppose that • The input function consists of a sum

If equals a sum? Suppose that • The input function consists of a sum of terms of the kind listed in the above table i. e. • The trial forms corresponding to be. Then, the particular solution of the given non-homogeneous differential equation is In other words, the form of is a linear combination of all the Linearly independent functions generated by repeated differentiation of the input function .

Example 1 Solve the non homogenous equation by using Undetermined Coefficient Method. Solution: For

Example 1 Solve the non homogenous equation by using Undetermined Coefficient Method. Solution: For complementary solution consider homogenous equation Auxiliary equation is

For particular solution we assume Substituting into the given differential equation Equating the coefficients

For particular solution we assume Substituting into the given differential equation Equating the coefficients of and constant we have Constant: Solving these equations we get the values of undetermined coefficients

Thus Hence the general solution is

Thus Hence the general solution is

Example 2 Solve the non homogenous equation by using Undetermined Coefficient Method. Solution:

Example 2 Solve the non homogenous equation by using Undetermined Coefficient Method. Solution:

Duplication between and ? • If a function in the assumed is also present

Duplication between and ? • If a function in the assumed is also present in then this function is a solution of the associated homogeneous differential equation. In this case the obvious assumption for the form of is not correct. • In this case we suppose that the input function is made up of terms of kinds i. e. and corresponding to this input function the assumed particular solution is

 • If a contain terms that duplicate terms in , then that must

• If a contain terms that duplicate terms in , then that must be multiplied with , being the least positive integer that eliminates the duplication. Example 3 Find a particular solution of the following non-homogeneous differential Equation Solution: To find , we solve the associated homogeneous differential equation We put in the given equation, so that the auxiliary equation is

Thus Since Therefore Substituting in the given non-homogeneous differential equation, we obtain or Clearly

Thus Since Therefore Substituting in the given non-homogeneous differential equation, we obtain or Clearly we have made a wrong assumption for remove the duplication. , as we did not

Since is present in. Therefore, it is a solution of the associated homogeneous differential

Since is present in. Therefore, it is a solution of the associated homogeneous differential equation To avoid this we find a particular solution of the form We notice that there is no duplication between Assumption for. Now and this new Substituting in the given differential equation, we obtain or

So that a particular solution of the given equation is given by Hence, the

So that a particular solution of the given equation is given by Hence, the general solution of the given equation is Example 4 Find a particular solution of Solution: Consider the associated homogeneous equation Put

Then the auxiliary equation is Roots of the auxiliary equation are real and equal.

Then the auxiliary equation is Roots of the auxiliary equation are real and equal. Therefore, Since Therefore, we assume that This assumption fails because of duplication between We multiply with . Therefore, we now assume and .

However, the duplication is still there. Therefore, we again multiply with and assume Since

However, the duplication is still there. Therefore, we again multiply with and assume Since there is no duplication, this is acceptable form of the trial Example 5 Solve the initial value problem Solution Associated homogeneous DE .

Put Then the auxiliary equation is The roots of the auxiliary equation are complex.

Put Then the auxiliary equation is The roots of the auxiliary equation are complex. Therefore, the complementary function is Since Therefore, we assume that

So that Clearly, there is duplication of the functions and. To remove this duplication

So that Clearly, there is duplication of the functions and. To remove this duplication we multiply with. Therefore, we assume that So that Substituting into the given non-homogeneous differential equation, we have Equating constant terms and coefficients of we obtain , ,

So that Thus Hence the general solution of the differential equation is We now

So that Thus Hence the general solution of the differential equation is We now apply the initial conditions to find Since Therefore and

Now Therefore Hence the solution of the initial value problem is Example 6 Solve

Now Therefore Hence the solution of the initial value problem is Example 6 Solve the differential equation

Solution: The associated homogeneous differential equation is Put Then the auxiliary equation is Thus

Solution: The associated homogeneous differential equation is Put Then the auxiliary equation is Thus the complementary function is Since We assume that Corresponding to:

Corresponding to: Thus the assumed form of the particular solution is The function in

Corresponding to: Thus the assumed form of the particular solution is The function in is duplicated between and. Multiplication with does not remove this duplication. However, if we multiply with , this duplication is removed. Thus the operative from of a particular solution is Then And Substituting into the given differential equation and collecting like term, we obtain

Equating constant terms and coefficients of and yields Solving these equations, we have the

Equating constant terms and coefficients of and yields Solving these equations, we have the values of the unknown coefficients Thus, . Hence, the general solution

Higher –Order Equation The method of undetermined coefficients can also be used for higher

Higher –Order Equation The method of undetermined coefficients can also be used for higher order equations of the form with constant coefficients. The only requirement is that of the proper kinds of functions as discussed earlier. Example 7 Solve Solution: To find the complementary function we solve the associated homogeneous differential equation consists

Put Then the auxiliary equation is Or The auxiliary equation has equal and distinct

Put Then the auxiliary equation is Or The auxiliary equation has equal and distinct real roots. Therefore, the complementary function is Since Therefore, we assume that Clearly, there is no duplication of terms between and .

Summary (Recall)

Summary (Recall)