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:

Secondly, This equation can be rewritten as:

Thirdly, Now, let us make the substitution . Then, the differential equation can be rewritten as:

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

Substituting , we get:

Simplifying, we get:

Dividing both sides by , we get:

Separating variables, we get:

Integrating both sides, we get:

Simplifying, we get:

Fifthly, Substituting back for u, we get:

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.

Useful Links