Solution of Ex:2.5 Differential Equations with Boundary-Value Problems (7th Edition Book)

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Solving differential equations with boundary-value problems can be a challenging task, but with the right tools and techniques, it can be made simpler. However, In the 7th edition book, “Differential Equations with Boundary-Value Problems,”. Therefore, one such technique is discussed for solution of Ex:2.5 Differential Equations with boundary-value problems using an appropriate substitution.

Lastly, Solutions of problems of Ex:2.5 Differential Equations

Example for Understanding

First of all, let us consider the differential equation:

    \[y'' + 2y' + y = 0\]

Secondly, This equation can be rewritten as:

    \[(y' + y)^2 = y^2\]

Thirdly, Now, let us make the substitution u = y' + y. Then, the differential equation can be rewritten as:

    \[u^2 = y^2\]

Fourthly, Differentiating both sides with respect to x, we get:

    \[2uu' = 2yy'\]

Substituting y' = u - y, we get:

    \[2uu' = 2y(u - y)\]

Simplifying, we get:

    \[2uu' = 2uy - 2y^2\]

Dividing both sides by 2y, we get:

    \[u'/u = 1/y - u/2y\]

Separating variables, we get:

    \[2ydu/u = (2 - u)dy/y\]

Integrating both sides, we get:

    \[2ln|u| = 2ln|y| - y + C\]

Simplifying, we get:

    \[u = Cye^(-y)\]

Fifthly, Substituting back for u, we get:

    \[y' + y = Cye^(-y)\]

This is a first-order linear differential equation, which can be solved using standard techniques such as integrating factors.

Short Description for solution of Ex:2.5 Differential Equations

Finally, the Ex:2.5 differential equation with boundary-value problems can be solved using an appropriate substitution. By making the substitution u = y’ + y and following the above steps, we were able to convert the second-order differential equation into a first-order linear differential equation. However, these can be solved using standard techniques. Finally, this technique demonstrates the power of appropriate substitutions in solving differential equations with boundary-value problems.

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