A particular solution of \(\log (d y / d x)=3 x+4 y, y(0)=0\) is: (A) \(3 \mathrm{e}^{3 \mathrm{x}}+4 \mathrm{e}^{4 \mathrm{y}}=7\) (B) \(4 \mathrm{e}^{3 \mathrm{x}}-\mathrm{e}^{-4 \mathrm{y}}=3\) (C) \(\mathrm{e}^{3 \mathrm{x}}+3 \mathrm{e}^{-4 \mathrm{y}}=4\) (D) \(4 \mathrm{e}^{3 \mathrm{x}}+3 \mathrm{e}^{-4 \mathrm{y}}=7\)

In many differential equation problems, the natural logarithm, denoted as \( \ln(x) \), plays a crucial role. It is the logarithm to the base \(e\), where \(e\) is the irrational constant approximately equal to 2.71828. The natural logarithm allows us to relate multiplicative changes and growth rates to linear representations, which is particularly useful when dealing with continuous growth or decay problems.

One of the key properties of the natural logarithm is that it is the inverse function of the exponential function with base \(e\). This means that \( e^{\ln(x)} = x \) and \( \ln(e^x) = x \), given \( x > 0 \). Applying this property, as seen in the step-by-step solution, allows us to go back and forth between the logarithmic and exponential forms, which is a very powerful technique in solving differential equations.

The ‘separation of variables’ method is a quintessential strategy for solving differential equations where the rates of change are separable. This method involves rearranging the equation to isolate the differentials of each variable on opposite sides of the equation. This process allows us to integrate and find functions for the dependent variables.

Practical Application

Looking at our example, you'll notice that once we removed the natural logarithm, the variables \(x\) and \(y\) were on different sides of the resulting equation, prompting us to separate them further before integrating. The separation of variables is particularly elegant because it simplifies complex relationships into a format that can be tackled using fundamental integration techniques.

Integration is a fundamental operation in calculus, closely associated with the concept of the area under a curve. In the context of differential equations, integration helps in finding a function when its rate of change, or derivative, is known.

Example Insight

The step-by-step solution demonstrates integration on both sides of the equation after separating the variables. The process involved substituting to make integration feasible and then integrating the exponentials on both sides. This step not only utilizes basic integration rules but also reveals the power of substitution to simplify integration, making the integral solvable. The constants of integration often play a role in defining a family of solutions, which is why the next step, applying initial conditions, is crucial.

An initial value problem in differential equations is a problem where you are not only asked to solve the differential equation but are also given a starting point, known as the initial condition. The initial condition allows us to solve for any arbitrary constants that arise during the integration process, yielding a specific solution instead of a general one.

Understanding the Role of Initial Conditions

During differentiation, information is lost about the original function, specifically constant terms. When we integrate to 'regain' the original function, the integration constants represent this lost information. The initial conditions help us recover this information and find the particular solution that applies to the specific scenario given. In the given step-by-step solution, the initial condition \(y(0)=0\) is used to find the value of the integration constant \(C\). The resulting value is then used to construct the particular solution to the differential equation.