Solution of Ex:4.4 Differential Equations with Boundary-Value Problems (7th Edition Book)

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Solution of Ex:4.4 Differential Equations. Undetermined coefficients and the superposition approach are powerful techniques for solving differential equations with boundary-value problems. These methods are particularly useful when dealing with second-order linear differential equations with constant coefficients.

The undetermined coefficients method is based on the idea that if a differential equation can be expressed as a linear combination of a known set of functions, then the solution can also be expressed as a linear combination of those functions. In other words, if we can find a particular solution to the differential equation, then the general solution can be expressed as the sum of the particular solution and the complementary solution.

Solution of Ex:4.4 Differential Equations with Boundary-Value Problems

To find a particular solution, we assume that it takes the same form as the forcing function, but with unknown coefficients. We then substitute this solution into the differential equation and solve for the coefficients. The complementary solution is found by solving the homogeneous equation associated with the differential equation.

EXAMPLE 01

consider the differential equation:

y” + 3y’ + 2y = 2x + 3e^{-x}

The homogeneous equation associated with this differential equation is:

y” + 3y’ + 2y = 0

The complementary solution to this equation is:

y_c(x) = c_1e^{-x} + c_2e^{-2x}

To find the particular solution, we assume that it has the form:

y_p(x) = Ax + Be^{-x} + Ce^{-2x} + De^{-x}

where A, B, C, and D are constants to be determined. We substitute this solution into the differential equation:

y”_p + 3y’_p + 2y_p = 2x + 3e^{-x}

and solve for the coefficients. In this case, we find that:

A = 1/2, B = -1/2, C = 0, D = 3/2

Therefore, the particular solution is:

y_p(x) = (1/2)x – (1/2)e^{-x} + (3/2)e^{-x}

The general solution to the differential equation is then:

y(x) = y_c(x) + y_p(x) = c_1e^{-x} + c_2e^{-2x} + (1/2)x – (1/2)e^{-x} + (3/2)e^{-x}

The superposition approach is a variation of the undetermined coefficients method that is used when the forcing function is a linear combination of known functions. In this case, we assume that the particular solution is also a linear combination of functions with unknown coefficients. We then substitute this solution into the differential equation and solve for the coefficients.

EXAMPLE 02

consider the differential equation:

y” + 3y’ + 2y = 4\sin(x) – 6\cos(x)

The complementary solution to this equation is the same as before:

y_c(x) = c_1e^{-x} + c_2e^{-2x}

To find the particular solution, we assume that it has the form:

y_p(x) = A\sin(x) + B\cos(x)

where A and B are constants to be determined. We substitute this solution into the differential equation:

y”_p + 3y’_p + 2y_p = 4\sin(x) – 6\cos(x)

and solve for the coefficients. In this case, we find that:

A = -2, B = -1

Therefore, the particular solution is:

y_p(x) = -2\sin(x) – \cos(x)

The general solution to the differential equation is then:

y(x) = y_c(x) + y_p(x) = c_1e^{-x} +c_2e^{-2x} – 2\sin(x) – \cos(x)

Conclusion of olution of Ex:4.4 Differential Equations

In conclusion, undetermined coefficients and the superposition approach are powerful techniques for solving differential equations with boundary-value problems. These methods can be used to find particular solutions and general solutions to differential equations with constant coefficients. By assuming a specific form of the particular solution and solving for unknown coefficients, these methods enable us to find a solution that satisfies the given boundary conditions.

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