Question

If the Wronskian $W$ of $f$ and $g$ is $3e^{4t}$, and if $f(t)=e^{2t},$ find $g(t)$

Short answer

Question: Determine the function g(t) given that the Wronskian W of the functions f(t) and g(t) is 3e^(4t) and the function f(t) is e^(2t). Answer: g(t) = 3te^(2t) + Ce^(2t), where C is an arbitrary constant.

Step-by-step solution

Write down the given information

We are given the following: - Wronskian, $W=3e^{4t}$ - Function $f(t)=e^{2t}$ Our task is to find the function $g(t)$.

Compute the derivative of $f(t)$

In order to use the Wronskian formula, we need to find the derivative of $f(t)$. Differentiating $f(t)=e^{2t}$ with respect to $t$, we get: $f^{\prime }(t)=2e^{2t}$

Write down the Wronskian formula and substitute the given information

We know that the Wronskian is given by the formula: $W(f,g)=f{g}^{\prime }-f^{\prime }g$ Substitute the given information into the formula: $3e^{4t}=(e^{2t})(g^{\prime }(t))-(2e^{2t})(g(t))$

Solve the differential equation for $g(t)$

Now, we need to solve the differential equation for $g(t)$. This is a first-order linear inhomogeneous differential equation. Let's first divide through by $e^{2t}$: $\frac{3e^{4t}}{e^{2t}}=g^{\prime }(t)-2g(t)$ Simplify the equation: $3e^{2t}=g^{\prime }(t)-2g(t)$ Let's write it in the form of a first-order linear differential equation: $g^{\prime }(t)-2g(t)=3e^{2t}$

Use the integrating factor method to solve the differential equation

We can solve this differential equation using the integrating factor method. First, we calculate the integrating factor ($IF$): $IF=e^{\int -2dt}=e^{-2t}$ Now, multiply both sides of the differential equation by the integrating factor: $e^{-2t}{g}^{\prime }(t)-2e^{-2t}g(t)=3e^{2t}{e}^{-2t}$ This simplifies to: $(e^{-2t}g(t){)}^{\prime }=3$ Now, integrate both sides with respect to $t$: $\int (e^{-2t}g(t){)}^{\prime }dt=\int 3dt$ This gives: $e^{-2t}g(t)=3t+C$ Now, multiply both sides by the integrating factor's inverse, which is $e^{2t}$: $g(t)=3t{e}^{2t}+C{e}^{2t}$ Here, $g(t)$ is the function we were looking for, and $C$ is an arbitrary constant.

Key concepts

Differential Equations

A differential equation is a mathematical equation that involves functions and their derivatives. These equations can describe various phenomena such as growth, decay, oscillation, and waves. Differential equations are categorized based on the order of the highest derivative and whether they are linear or non-linear.

• The **order** of a differential equation is determined by the highest derivative present. For example, a second-order differential equation has the second derivative but not higher.
• A **linear** differential equation has solutions that form a straight line, while non-linear ones do not.

Understanding differential equations is fundamental in fields such as physics, engineering, and biology, as they often model real-world problems. In our exercise, we deal with a first-order linear differential equation, which is one of the simplest forms and makes use of an integrating factor to find solutions.

Integrating Factor

The integrating factor is a mathematical tool used to solve linear differential equations. This method is particularly useful for equations that are not directly integrable.

• To find the integrating factor ($IF$, we calculate $e^{\int P(t)dt}$ where $P(t)$ is the function multiplying $y$ in $y^{\prime }(t)+P(t)y=Q(t)$.
• The goal of the integrating factor is to turn the differential equation into an exact differential, which can then be integrated straightforwardly.

For our specific problem, the integrating factor was $e^{-2t}$, calculated from the given differential equation $g^{\prime }(t)-2g(t)=3e^{2t}$. Multiplying the entire equation by this factor transforms it, making it easier to solve.

Inhomogeneous Differential Equation

An **inhomogeneous differential equation** is one that contains a non-zero term independent of the function being solved for. This term is called the forcing function or non-homogeneous term and adds extra complexity to the equation.

• A simple form of an inhomogeneous differential equation looks like $y^{\prime }(t)+P(t)y=Q(t)$, where $Q(t)$ is not equal to zero.
• Finding solutions often involves both a complementary (homogeneous) solution and a particular solution that addresses the non-homogeneous part.

In our case, the term $3e^{2t}$ is the non-homogeneous part, requiring special methods like the integrating factor to find a complete solution.

Function Derivation

Function derivation is the process of finding the derivative of a function, which measures how the function changes as its input changes. The derivative is a fundamental concept in calculus and is symbolized by $f^{\prime }(t)$ or $dy/dt$.

• Derivatives are used to determine slopes of curves, rates of change, and to solve differential equations.
• The process involves rules and techniques such as the product rule, chain rule, and power rule.

In our exercise, the derivation of $f(t)=e^{2t}$ leads to $f^{\prime }(t)=2e^{2t}$, which is essential for applying the Wronskian and solving the differential equation for $g(t)$. Function derivation allows for a more in-depth analysis of how functions behave and interact over different values of $t$.