Problem 1 - Differential Equations and Linear Algebra

Question

Solve the given differential equation. $$\frac{d y}{d x}=2 x y$$

Step-by-Step Solution

Answer

The general solution to the given differential equation is \(y(x) = \pm C_2 e^{x^2}\), where \(C_2\) is a constant.

Step 1: Separate Variables

First, we need to separate the variables in the given equation. We are given, \(\frac{dy}{dx} = 2xy\). Divide both sides by \(y\) and multiply both sides by \(dx\) to separate the variables, we get \[\frac{dy}{y} = 2x dx.\]

Step 2: Integrate Both Sides

Now, integrate both sides of the equation. \[\int \frac{dy}{y} = \int 2x dx.\] On the left side, we have the natural logarithm of the absolute value of \(y\), and on the right side, the integral of \(2x\) is \(x^2\) plus a constant of integration, which we can denote as \(C_1\). So, we have: \[\ln |y| = x^2 + C_1.\] To remove the natural logarithm, we can exponentiate both sides of the equation.

Step 3: Exponentiate Both Sides

Exponentiate both sides, we get: \[|y| = e^{x^2 + C_1}.\] Here, we can rewrite the right side as \[|y| = e^{x^2} \cdot e^{C_1}.\] Since \(e^{C_1}\) is also a constant, we can represent it as a new constant, say \(C_2\). So, the equation becomes: \[|y| = C_2 e^{x^2}.\]

Step 4: Solve for y

Since the absolute value of \(y\) is equal to a constant times \(e^{x^2}\), we can express \(y\) as the following: \[y(x) = \pm C_2 e^{x^2}.\] Here, the function y(x) is the general solution to the given differential equation.

Key Concepts

Separation of Variables

In a differential equation, separating variables is the essential first step when the variables are mixed together. Our goal is to rewrite the equation so that each variable is on a different side. Consider the equation \( \frac{dy}{dx} = 2xy \). To separate the variables, we manipulate the equation:

• Divide both sides by \( y \) to isolate \( dy \) on one side.
• Multiply both sides by \( dx \) to shift \( dx \) to the opposite side.

This results in \( \frac{dy}{y} = 2x \, dx \), effectively separating \( y \) and \( x \). Each part now only contains one of the two variables, simplifying further calculations.

Integration

Once the variables are separated, integration allows us to solve the equation. We have:\[ \int \frac{dy}{y} = \int 2x \, dx. \]

• The left side integrates as the natural logarithm: \( \ln |y| \).
• The right side integrates to \( x^2 \), remembering to add a constant of integration, \( C_1 \).

Thus, we arrive at: \( \ln |y| = x^2 + C_1 \). Adding constants is crucial, as each represents a different possible solution to the differential equation, representing the general solution.

Exponentiation

Exponentiating both sides resolves the logarithm, reverting back to \( y \). From the equation \( \ln |y| = x^2 + C_1 \), we can raise \( e \) to the power of both sides:

• The result is \( |y| = e^{x^2 + C_1} \).
• Rewriting \( e^{x^2 + C_1} \) as \( e^{x^2} \cdot e^{C_1} \) simplifies further to \( |y| = C_2 e^{x^2} \), where \( C_2 \) is \( e^{C_1} \).

This step helps simplify the expression, transforming the solution back into a more recognizable form, isolated for \( y \).

General Solution

The general solution of the differential equation embodies all possible solutions considering an arbitrary constant. From \( |y| = C_2 e^{x^2} \), we know that since \( |y| \) can be positive or negative, this translates to:

• \( y(x) = \pm C_2 e^{x^2} \)
• The \( \pm \) accounts for both positive and negative solutions for \( y \).

This function \( y(x) \) represents a family of solutions. The inclusion of the constant \( C_2 \) highlights that for different constants, there are different, specific solutions representing the general behavior of \( y \).