Question

What is the general solution of the equation $y^{\prime }(t)=3y-12?$

Short answer

Answer: The general solution of the given ODE is $y(t)=4+C{e}^{3t}$, where C is the constant of integration.

Step-by-step solution

Identify the ODE and its form

The given equation is a first-order linear ordinary differential equation (ODE) in the form: $$y^{\prime }(t)-3y(t)=-12$$

Find the integrating factor

The integrating factor (IF) is given by the formula: $$IF=e^{\int P(t)dt}$$ Here, P(t) is the coefficient of y(t) which is -3 in this case. So, $$IF=e^{\int -3dt}=e^{-3t}$$

Multiply the ODE by the integrating factor

Multiply the given ODE by the integrating factor, so we get: $$(e^{-3t}{y}^{\prime }(t)-3e^{-3t}y(t))=-12e^{-3t}$$ We can rewrite the left-hand side as the derivative of a product: $$\frac{d}{dt}(e^{-3t}y(t))=-12e^{-3t}$$

Integrate the resulting equation

Now, we find the integral of both sides concerning t: $$\int \frac{d}{dt}(e^{-3t}y(t))dt=\int -12e^{-3t}dt$$ This simplifies to: $$e^{-3t}y(t)=4e^{-3t}+C$$, where C is the constant of integration.

Solve for the unknown function y(t)

To find the general solution, y(t), we will isolate y(t) in the equation. By dividing both sides by $e^{-3t}$, we obtain: $$y(t)=4+C{e}^{3t}$$ This is the general solution of the given ODE.

Key concepts

Integrating Factor

An integrating factor is a powerful tool used to solve first-order linear ordinary differential equations (ODEs). It's essentially a function that, when multiplied by the ODE, simplifies the equation by turning it into an exact differential, allowing for straightforward integration. How do we find this magical function? It derives from the formula $IF=e^{\int P(t)dt}$, where $P(t)$ is the coefficient of the function $y(t)$ in the ODE.

Let's take the exercise $y^{\prime }(t)=3y-12$ and apply this concept. By identifying $P(t)=-3$ and integrating, we obtain the integrating factor $e^{-3t}$ which is applied to every term of the ODE. The resulting equation $e^{-3t}{y}^{\prime }(t)-3e^{-3t}y(t)=-12e^{-3t}$ showcases that the left-hand side is the derivative of the product $e^{-3t}y(t)$ which can then be integrated to further solve the equation. The beauty of the integrating factor lies in its simplicity—transforming complex differential equations into more manageable forms.

First-Order Linear ODE

A first-order linear ODE is a differential equation that involves only the first derivative of the unknown function and linear terms of the function itself. The general form can be expressed as $y^{\prime }+P(t)y=Q(t)$ where $y^{\prime }$ represents the first derivative of $y$, and $P(t)$ and $Q(t)$ are functions of $t$. This specific type of ODE is known for its trait of being directly solvable by using an integrating factor.

From the exercise, we have the equation $y^{\text{'}prime}(t)=3y-12$, a textbook example of a first-order linear ODE where $P(t)=-3$ and $Q(t)=-12$ are constants. Such constants make the application of the integrating factor straightforward. The process normally involves obtaining the integrating factor, multiplying the entire ODE by it, and then integrating to find the solution—ways these steps create a clear pathway from a seemingly daunting differential equation to a neat, generalized solution.

General Solution of ODE

The general solution of an ODE encapsulates all the particular solutions and includes arbitrary constants representing the infinite variety of solutions that exist due to initial conditions. For first-order linear ODEs, after applying the integrating factor and simplifying the equation, one integrates to find an expression involving the unknown function and an integration constant.

In the exercise given, we end up with $e^{-3t}y(t)=4e^{-3t}+C$ post integration. Solving for $y(t)$ involves ridding the equation of the integrating factor by dividing, which leaves us with $y(t)=4+C{e}^{3t}$—the general solution. Here, $C$ represents an infinite number of possibilities that depend on initial values or boundary conditions. The concept of a general solution is crucial as it represents a complete set of solutions from which one can derive specific solutions through additional information, such as initial values. It is core to not just understanding but also to applying ODEs to real-world problems where conditions vary widely.