Problem 36 - Calculus and its applications

Question

Solve by separating variables. \(\frac{d y}{d x}=5 x^{4} y\)

Step-by-Step Solution

Answer

The solution is \( y = Ce^{x^{5}} \), where \( C \) is a constant.

Step 1: Rewrite the equation

Start by rewriting the differential equation \( \frac{d y}{d x}=5 x^{4} y \) so that all \( y \) terms and their differentials are on one side, and all \( x \) terms and their differentials are on the other side. This gives us \( \frac{d y}{y}=5 x^{4} d x \).

Step 2: Integrate both sides

Integrate both sides of the separated equation. The left side integrates to \( \int \frac{1}{y} \, dy = \ln |y| \). The right side integrates to \( \int 5 x^{4} \, dx = x^{5} + C \), where \( C \) is the constant of integration.

Step 3: Solve for y

Raise both sides of the equation \( \ln |y| = x^{5} + C \) by using the exponent function to solve for \( y \). This gives \( y = e^{x^{5} + C} \). Notice that \( e^{C} \) is still just a constant, so we can rewrite this as \( y = Ce^{x^{5}} \).

Key Concepts

Separation of Variables

Separation of Variables is a powerful technique used to solve ordinary differential equations (ODEs), specifically when the equation can be expressed in a form where all terms involving one variable (and its differential) are grouped on one side, and all terms involving the other variable are on the opposite side.
This "separation" paves the way for straightforward integration of each side.

To use separation of variables, start by rewriting the ODE such that one side solely consists of terms with one variable, say \(y\), while the terms with the other variable, say \(x\), reside on the opposite side.
For instance, given the equation \(\frac{dy}{dx} = 5x^4y\), we separate variables by manipulating it to \(\frac{dy}{y} = 5x^4 \, dx\).
This sets the equation in a form ready for integration.

• All terms with \(y\) and \(dy\) go together.
• All terms with \(x\) and \(dx\) go together.

Step by step, this method breaks down complex equations into more manageable parts, eventually leading to a solution for the dependent variable, \(y\).
It's particularly useful when the direct integration of separate parts is possible.

Integration

Integration is the process of finding the integral of a function, a central operation in solving differential equations through separation of variables.
In the context of our differential equation, after separating the variables, we integrate both sides to solve for the function \(y\).

The left side of the separated equation \(\int \frac{1}{y} \, dy\) simplifies using the natural logarithm, resulting in \(\ln |y|\).
This is a common integration result when dealing with inverses of linear functions.

• The integral of \(\frac{1}{y}\) is a logarithmic function.
• The absolute value in \(\ln |y|\) ensures the logarithm outcome remains valid for negative \(y\) values.

On the right side, \(\int 5x^4 \, dx\) involves a standard power rule integration, resulting in \(x^5\), plus a constant \(C\).
Power rule integration is systematic; simply increase the power by one and divide by the new exponent.
Additionally, don't forget to add the integration constant, as it represents the general solution's family.
Understanding how to integrate different types of functions broadens the scope of solvable equations using this method.

Exponential Functions

Exponential functions play a crucial role in solutions to differential equations, especially when the differential equation involves separated variables.
In our exercise, after integration, the equation \(\ln |y| = x^5 + C\) involves a logarithmic relationship which leads us naturally to use exponentials.

To solve for \(y\), we utilize properties of exponents by employing the inverse operation to logarithms.
This gives \(e^{x^5 + C}\).

• The inverse of natural logarithm \((\ln)\) is the exponential function \(e\).
• The result, \(e^{x^5 + C}\), simplifies to \(y = Ce^{x^5}\), where \(C\) is a constant.

This simplification uses the fact that \(e^{C}\) can be viewed as a constant, \(C\), since it's fixed across all solutions in this equation's general form.
Exponential functions describe rapid growth or decay in many physical phenomena, making them a fundamental aspect in the behavior of solutions to such differential equations.