Non Homogeneous Linear Equations with Constant Coefficients

TL;DR

Learn how to solve non-homogeneous linear partial differential equations step-by-step.

Transcript

click the Bell icon to get latest videos from akira hello friends in the chapter of partial differential equation we have seen various types in the last video we saw how to solve homogeneous linear partial differential equations in this video we will see how to solve non homogeneous linear partial differential equations after seeing partial differe... Read More

Key Insights

• ✊ Non-homogeneous linear partial differential equations can have various powers of partial derivatives.
• 🥳 The solution for these equations consists of two parts: the complementary function and the particular integral.
• 🔚 Factorization of the function of D and D' is required to find the complementary function.
• ❓ The particular integral is found by using the same method as for homogeneous equations.
• ❓ The final solution is obtained by combining the complementary function and particular integral.
• 🔚 Understanding the properties and rules of D and D' operators is crucial for solving these equations.
• 0️⃣ The derivative of a constant is always zero.

Questions & Answers

Q: What is the main difference between homogeneous and non-homogeneous linear partial differential equations?

The main difference is that non-homogeneous equations can have various powers of partial derivatives, while homogeneous equations only have zeroth, first, or second order derivatives of the function.

Q: How is the complementary function of non-homogeneous linear partial differential equations obtained?

The complementary function is obtained by factorizing the function of D and D' into exponential and polynomial terms and then combining them in a specific format.

Q: What is the particular integral and how is it found for non-homogeneous equations?

The particular integral is one part of the final solution and it is found by applying the same method used for homogeneous equations. It involves finding the function of D and D' that satisfies the given non-homogeneous equation.

Q: Can you provide an example of solving a non-homogeneous linear partial differential equation?

Yes, let's consider the example where the function of D and D' is (B - D - 1)(D - D' - 2). We can compare it with the standard form and then find the complementary function and particular integral step-by-step.

Summary & Key Takeaways

• Non-homogeneous linear partial differential equations include various powers of partial derivatives.

• To solve these equations, factorize the function of D and D' and find the complementary function and particular integral.

• The complementary function is a combination of exponential and polynomial terms, while the particular integral is found using the same method as homogeneous equations.