Question

Solving initial value problems Solve the following initial value problems. $$u^{\prime \prime}(x)=4 e^{2 x}-8 e^{-2 x}, u(0)=1, u^{\prime}(0)=3$$

Short answer

Answer: The specific solution is \(u(x) = -x + \frac{3}{2} + \frac{1}{2}e^{2x} - e^{-2x}\).

Step-by-step solution

Find the general solution to the homogeneous equation

To find the general solution of the homogeneous equation, let's first set: \(u''(x) = 0\). The resulting homogeneous equation is: $$u^{\prime \prime}(x)=0$$ The general solution is given by a linear combination of the independent solutions \(e^{mx}\), where \(m\) is the exponent. Since we have a second-order equation, we need two independent solutions. The general solution for this homogeneous equation is a combination of a first and a second-degree polynomials: $$u_h(x) = Ax + B$$ where A and B are constants.

Find a particular solution to the inhomogeneous equation

To find a particular solution to the inhomogeneous equation, we will use the method of undetermined coefficients. We are given: $$u^{\prime \prime}(x)=4 e^{2 x}-8 e^{-2 x}$$ We need to guess a particular solution that has the same form as the right-hand side, but not the same form as the homogeneous solution. We guess: $$u_p(x) = Ce^{2x} + De^{-2x}$$ Now we need to find the first and the second derivatives of our guessed particular solution: $$u_p'(x) = 4Ce^{2x} - 4De^{-2x}$$ $$u_p''(x) = 8Ce^{2x} + 8De^{-2x}$$ Now, by substituting the derivatives of the particular solution into the given differential equation, we can determine the constants C and D: $$8Ce^{2x} + 8De^{-2x} = 4e^{2x} - 8e^{-2x}$$ This equation implies that \(C = \frac{1}{2}\) and \(D = -1\). Therefore, the particular solution to the inhomogeneous equation is: $$u_p(x) = \frac{1}{2}e^{2x} - e^{-2x}$$

Combine the homogeneous and particular solutions

To find the general solution to the given differential equation, we need to combine the homogeneous solution \(u_h(x)\) and the particular solution \(u_p(x)\): $$u(x) = u_h(x) + u_p(x) = Ax + B + \frac{1}{2}e^{2x} - e^{-2x}$$

Use the initial conditions to find the specific solution

We are given the initial conditions \(u(0) = 1\) and \(u'(0) = 3\), which we use to solve for the constants A and B: $$u(0) = A(0) + B + \frac{1}{2}e^{2(0)} - e^{-2(0)} = 1$$ $$1 = B - \frac{1}{2}$$ Thus, \(B = \frac{3}{2}\). Now, we will find \(u'(x)\): $$u'(x) = A + 2e^{2x} + 2e^{-2x}$$ We can now use the second initial condition \(u'(0) = 3\): $$u'(0) = A + 2e^{2(0)} + 2e^{-2(0)} = 3$$ $$A + 4 = 3$$ Thus, \(A = -1\).

Write the final specific solution

Combining all the previous steps, we find the specific solution to the given differential equation, with the initial conditions: $$u(x) = -x + \frac{3}{2} + \frac{1}{2}e^{2x} - e^{-2x}$$

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They describe how a particular quantity changes in relation to another. Understanding differential equations is crucial because they can model real-world phenomena like population growth, heat transfer, or motion dynamics.

There are primary types of differential equations: ordinary differential equations (ODEs) and partial differential equations (PDEs). Ordinary differential equations involve functions of one variable and their derivatives. Expand your knowledge by exploring different methods of solving these equations as these principles can extend to various applications, such as engineering or physics problems.

Homogeneous Equation

A homogeneous equation in the context of differential equations has the form where all terms depend on the dependent variable and its derivatives. A standard homogeneous differential equation might be represented as \( a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \, ... \, + a_1(x)y' + a_0(x)y = 0 \).

In other words, there is no separate function on the right side of the equation. For solving homogeneous equations, we often look for solutions that can be expressed as exponential functions, polynomials, or sine-cosine forms, depending on the order and type of the equation.

• Linear homogeneous differential equations have constant coefficients and can be solved using methods involving characteristic equations.
• Understanding the solution to the homogeneous part is crucial as it forms the basis of solving the entire differential equation.

When dealing with higher-order equations, always ensure to calculate each independent solution. This step is crucial when preparing to solve more complex inhomogeneous equations.

Particular Solution

Finding a particular solution is essential when solving inhomogeneous differential equations, which include non-zero terms independent of the dependent variables and their derivatives. The general form of such an equation can be written as \( a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \, ... \, + a_1(x)y' + a_0(x)y = g(x) \), where \(g(x) eq 0\).

The particular solution, noted as \(u_p(x)\), satisfies the differential equation but not necessarily any initial conditions.

• An effective method to find it is through guessing a form similar to \(g(x)\) that doesn't conflict with existing terms in the homogeneous equation.
• Once guessed, substitute this form into the original differential equation to solve for any unknown coefficients.

Understanding this concept will solidify your ability to solve complex differential equations by decoupling the homogeneous and non-homogeneous parts.

Method of Undetermined Coefficients

The method of undetermined coefficients is a technique used to find a particular solution to a non-homogeneous linear differential equation with constant coefficients. When the non-homogeneous part \( g(x) \) of the differential equation has a form such as a polynomial, exponential, sine, or cosine function, this method can be particularly handy.

Here's a simple approach to use this method:

• Identify the form of \(g(x)\) in the differential equation.
• Guess a form for the particular solution \( u_p(x) \) that reflects the form of \(g(x)\).
• Differentiating this guess, substitute it back into the original equation.
• Compare coefficients to determine any constant values, thereby finding \(u_p(x)\).

This tool is impactful because it relies not on solving complex integrations, but on intelligent guessing and applying patterns. Once you recognize these patterns, solving similar equations becomes more intuitive, aiding in broader problem-solving scenarios in mathematics and physics.