Question

For the given differential equation, (a) Determine the complementary solution, \(y_{c}(t)=c_{1} y_{1}(t)+c_{2} y_{2}(t)\). (b) Use the method of variation of parameters to construct a particular solution. Then form the general solution. \(y^{\prime \prime}-[2+(1 / t)] y^{\prime}+[1+(1 / t)] y=t e^{t}, \quad 0

Short answer

Answer: The specific solution is \(y(x) = -\frac{1}{2} \cos(2x) - 2 \sin(2x)\).

Step-by-step solution

Identify the auxiliary equation

The given differential equation is: $$ y^{\prime\prime} + 4y = 0 $$ From this equation, we can write the auxiliary equation as: $$ m^2 + 4 = 0 $$

Solve the auxiliary equation

To find the roots of the auxiliary equation, solve for m: $$ m^2 = -4 $$ So, m will be: $$ m = \pm 2i $$

Determine the general solution

Since the roots of the auxiliary equation are complex, the general solution will be in the form: $$ y(x) = C_1 \cos(2x) + C_2 \sin(2x) $$ where \(C_1\) and \(C_2\) are constants.

Apply the initial conditions

We have the initial conditions: $$ y\left(\frac{\pi}{4}\right) = -2, \quad y^{\prime}\left(\frac{\pi}{4}\right) = 1 $$ Substitute the initial conditions into the general solution and its derivative to find the constants \(C_1\) and \(C_2\): (a) For \(y\left(\frac{\pi}{4}\right) = -2\): $$ -2 = C_1 \cos\left(\frac{\pi}{2}\right) + C_2 \sin\left(\frac{\pi}{2}\right) $$ Since \(\cos(\frac{\pi}{2}) = 0\) and \(\sin(\frac{\pi}{2}) = 1\), this equation becomes: $$ -2 = C_2 $$ (b) For \(y^{\prime}\left(\frac{\pi}{4}\right) = 1\): First, find the derivative of the general solution: $$ y^{\prime}(x) = -2C_1 \sin(2x) + 2C_2 \cos(2x) $$ Now, substitute the initial condition: $$ 1 = -2C_1 \sin\left(\frac{\pi}{2}\right) + 2C_2 \cos\left(\frac{\pi}{2}\right) $$ Since \(\sin(\frac{\pi}{2}) = 1\) and \(\cos(\frac{\pi}{2}) = 0\), this equation becomes: $$ 1 = -2C_1 $$ So, \(C_1 = -\frac{1}{2}\).

Find the specific solution

Now that we have found \(C_1\) and \(C_2\), we can substitute them back into the general solution to find the specific solution: $$ y(x) = -\frac{1}{2} \cos(2x) - 2 \sin(2x) $$ This is the initial value problem's solution that satisfies the given initial conditions.

Key concepts

Complementary Solution

The complementary solution, often represented as \(y_c(t)\), is one of the core aspects of solving second-order linear homogeneous differential equations. It is constructed by finding a set of functions \(y_1(t)\) and \(y_2(t)\) that individually satisfy the homogeneous differential equation without the nonhomogeneous term (in this case, the term \(te^t\)). These functions are then used to construct the complementary solution as a linear combination of \(y_1(t)\) and \(y_2(t)\), expressed as \(y_c(t) = c_1 y_1(t) + c_2 y_2(t)\), where \(c_1\) and \(c_2\) are constants.

For the given differential equation, the complementary solution is made up of the solutions \(e^t\) and \(t^2 e^t\), as they are solutions to the corresponding homogeneous equation. By putting them together, students can find the general form of the complimentary solution, which serves as the basis for the differential equation's general solution. However, to fully determine the complementary solution, initial or boundary conditions must be applied.

Variation of Parameters

Variation of parameters is a method used to find a particular solution to nonhomogeneous differential equations. This technique is particularly useful when the nonhomogeneous term, such as \(te^t\) in our exercise, does not lend itself to simpler methods like undetermined coefficients.

To apply variation of parameters, we first need the complementary solution, which we have already discussed. Then, we assume that the constants \(c_1\) and \(c_2\) in the complementary solution are functions of \(t\) rather than mere constants. By substituting these functions into the original differential equation and using Wronskian determinants, we can derive formulas to determine \(c_1(t)\) and \(c_2(t)\). Combining these functions with our complementary solution functions \(y_1(t)\) and \(y_2(t)\), we obtain our particular solution. Finally, adding the particular solution to the complementary solution yields the general solution of the nonhomogeneous differential equation, which is the ultimate goal of the problem.

Auxiliary Equation

The auxiliary equation, sometimes also called the characteristic equation, is a pivotal tool for determining the complementary solution to a differential equation. It's derived from the homogeneous differential equation by assuming a solution of the form \(e^{mt}\), where \(m\) is a number that needs to be determined.

In our step by step solution example, the auxiliary equation \(m^2 + 4 = 0\) helps determine the nature of the complementary solution. The roots of the auxiliary equation here are complex, leading to a complementary solution composed of sine and cosine functions. Specifically, for our auxiliary equation, since the roots are \(\pm 2i\), the complementary solution is constructed using cosine and sine as \(y(x) = C_1 \cos(2x) + C_2 \sin(2x)\), showcasing how different types of roots (real, repeated, or complex) influence the form of the complementary solution.

When solving differential equations, it is essential to construct the auxiliary equation accurately as it sets the foundation for the rest of the solution process.