Higher Order Differential Equation when R.H.S = sinax,cosax - Problem 2

TL;DR

Learn how to solve a higher order differential equation with trigonometric functions as the right-hand side.

Transcript

friends so after completing the question number one let's move to the question number two where we'll see how to solve the problem on higher order differential equation when right hand side is sine ax or cos x so here we have to solve cosec x b raise to four y by d x raise to four plus y cosine x is equal to sine to it so as we have to solve this e... Read More

Key Insights

• ✋ Higher order differential equations with trigonometric functions as the right-hand side require transformation into a standard form for solving.
• 😄 The d-factorization formula of trigonometry is used to manipulate the equation and obtain the auxiliary equation.
• 🫚 The roots of the auxiliary equation determine the complex and distinct roots for the complementary function.
• 🍉 The particular integral is calculated by applying the operator 1/(d^4 + 1) on the trigonometric terms.
• 🍹 The final solution is the sum of the complementary function and the particular integral.
• 🅰️ Understanding the formulas and techniques from algebra and trigonometry is crucial for solving these types of differential equations.
• 🎮 Subscribing to ekeeda.com provides access to more videos and resources on engineering mathematics.

Questions & Answers

Q: How is the higher order differential equation with sine and cosine terms transformed into a standard form?

The equation is divided throughout with cosec x to cancel out the trigonometric functions and bring the coefficient of the higher order term to one.

Q: Why do we need to use the d-factorization formula of trigonometry?

We need to use the d-factorization formula to ensure that there is a plus or minus sign between any two trigonometric functions, allowing us to apply operators on either side of the equation.

Q: How are the roots of the auxiliary equation obtained in this case?

By treating the equation as a quadratic equation in d^2 and manipulating it using the formula for expanding (a+b)^2, we can find the roots of the equation.

Q: What is the formula for the complementary function with complex and distinct roots?

The complementary function is given by e^(real part * x) * (c1 * cos(imaginary part * x) + c2 * sin(imaginary part * x)).

Q: How is the particular integral calculated for this equation?

The particular integral is obtained by applying the 1/(d^4 + 1) operator on the right-hand side of the equation, considering the trigonometric functions as negative squares of a constant.

Q: What is the final solution for the higher order differential equation?

The final solution is the sum of the complementary function and the particular integral.

Summary & Key Takeaways

• The video focuses on solving a higher order differential equation with sine and cosine terms.

• The equation is transformed into a standard form by dividing throughout with cosec x.

• The auxiliary equation is found by equating the function of d to zero and solving for the roots.

• The solution includes a complementary function and a particular integral.