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Chapter 2: Problem 2

Find the general solution: (a) \(u_{x x}-2 u_{x y} \sin x-u_{y y} \cos ^{2} x-u_{y} \cos x=0\) (b) \(y^{2} u_{x x}-2 y u_{x y}+u_{y y}=u_{x}+6 y\)

Short Answer

Expert verified
General solutions for given equations can be found using method of separation of variables. The level of complexity will determine the number of steps but in each case, you'll simplify and reduce the equation to a form of simple differential equations that them can be solved.

Step by step solution

01

- Solve Equation A

Use the method of separation of variables where \( u(X, Y) = X(x)Y(y) \), substitute it into the equation and then simplify.
02

- General Solution for Equation A

After simplifying, the general solution for the equation \( u_{x x}-2 u_{x y} \sin x-u_{y y} \cos ^{2} x-u_{y} \cos x=0 \) can be found by reducing it to a form of a simplest differential equations that can be solved, such as a harmonic oscillator equation.
03

- Solve Equation B

Similar to step 1, input the solution form \( u(X, Y) = X(x)Y(y) \) into the equation and simplify.
04

- General Solution for Equation B

The general solution for the equation \( y^{2} u_{x x}-2 y u_{x y}+u_{y y}=u_{x}+6 y \) can be similarly obtained by reducing this equation to a form of simple differential equation, and solving the separated parts individually, then multiply them together to get the final result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Separation of Variables
The method of Separation of Variables is a crucial technique when solving partial differential equations (PDEs). Let鈥檚 break it down. In essence, this method allows you to break down a complicated PDE into simpler, ordinary differential equations (ODEs). Here鈥檚 how it works:
  • First, you assume a solution of the form \( u(x,y) = X(x)Y(y) \), where \(X\) depends only on \(x\) and \(Y\) depends only on \(y\).
  • By substituting this assumed form into the given PDE, you separate the variables such that each side of the equation relies solely on one of the variables.
  • This separation typically yields two separate ODEs, ideally much simpler to solve.
This technique is not only systematic but powerful as it transforms a complex multidimensional problem into solvable chunks. In the given exercise, applying separation of variables was key to initially tackle and simplify both equations A and B.
Harmonic Oscillator
In mathematical physics, the harmonic oscillator is a prominent concept. Within the realm of differential equations, especially when simplifying and solving equations, it often makes an appearance. But what is it in the simplest terms?
  • A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement.
  • In mathematical terms, it is often expressed by the equation \( \, \frac{d^2u}{dx^2} + ku = 0 \, \), where \(k\) represents a constant.
  • This equation鈥檚 solutions are generally sinusoidal functions, often sines and cosines.
Harmonic oscillators show up in physics as models of many real-world systems, like springs and pendulums. Solving parts of PDEs by reducing them to a harmonic oscillator equation turns the complexity of solving into something well understood. In our exercise, finding solutions akin to harmonic oscillators simplified tackling the equations effectively.
General Solution
The General Solution of a PDE is an expression encompassing all possible solutions. For any given PDE, obtaining the general solution is often the ultimate goal.
  • When solving a PDE, the general solution incorporates all possible functions that satisfy the equation.
  • It involves arbitrary constants or functions, representing the solution family.
  • By implementing initial or boundary conditions, these constants or functions can be precisely determined to find a specific solution.
In the exercise, after employing separation of variables, simplifying through harmonic oscillators, the general solutions for both equation A and B were found by addressing each reduced differential equation. The completeness of these solutions helps ensure that every possible configuration satisfying the equations in various contexts has been covered, demonstrating the power and utility of systematic approaches in solving PDEs.

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