Problem 4 - Calculus Early Transcendentals

Question

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \left(1+x^{4}\right) \frac{d y}{d x}=\frac{x^{3}}{y} $$

Step-by-Step Solution

Answer

The solution is \( y = \sqrt{\frac{1+x^4}{2} + C} \).

Step 1: Rewrite the Differential Equation

Start by rewriting the given differential equation \( (1 + x^4) \frac{dy}{dx} = \frac{x^3}{y} \) in the differential form. Multiply both sides by \( y \) to obtain \( y(1 + x^4) \frac{dy}{dx} = x^3 \).

Step 2: Separate Variables

To use the method of separation of variables, rearrange the equation from Step 1. Division by \((1 + x^4)\) and multiplication by \( dx \) gives \( y \, dy = \frac{x^3}{1 + x^4} \, dx \).

Step 3: Integrate Both Sides

Integrate both sides separately. The left side integrates to \( \frac{1}{2}y^2 + C_1 \) and the right side requires substitution \( u = 1 + x^4 \), resulting in \( \int \frac{x^3}{1 + x^4} \, dx = \int \frac{1}{4} \, du \), which integrates to \( \frac{1}{4}u + C_2 = \frac{1}{4}(1 + x^4) + C_2 \).

Step 4: Combine and Solve for y

Combine the integrated equations from Step 3 to \( \frac{1}{2}y^2 = \frac{1}{4}(1 + x^4) + C \), where \( C = C_2 - C_1 \). Solve for \( y \) to find \( y = \sqrt{\frac{1 + x^4}{2} + C} \), which is the explicit family of solutions.

Key Concepts

Separation of Variables

The method of separation of variables is a powerful technique used to solve certain types of differential equations. If you have an equation where the rate of change of a function is dependent on both the variable and the function itself, this method might come in handy. For example, consider the differential equation we had, \((1+x^{4}) \frac{dy}{dx}=\frac{x^{3}}{y}\).
To apply separation of variables, you need to rearrange the equation to isolate all terms involving \(y\) on one side and all terms involving \(x\) on the other. This leads to steps like multiplying or dividing both sides by a function of a variable or moving terms across the equation to separate them out.
For the given equation, after manipulating through multiplication and division as necessary, you obtain:

• \(y \, dy = \frac{x^3}{1 + x^4} \, dx\)

This indicates that the variables are now separated, paving the way for integration.

Integration

Integration is the process of finding the antiderivative or the area under the curve. In the context of solving differential equations, integration helps you move from the rate of change to the original function. Once the variables are separated, the next step is to integrate both sides of the equation independently.
Let's take our separated equation \(y \, dy = \frac{x^3}{1 + x^4} \, dx\). When integrating both sides, you need to treat \(y\) and \(x\) as independent variables, each subject to their own integration process:

• The left side, \(\int y \, dy\), integrates to \(\frac{1}{2}y^2 + C_1\).
• The right side, \(\int \frac{x^3}{1 + x^4} \, dx\), may seem tricky. A substitution such as \(u = 1 + x^4\) simplifies it, allowing you to transform and integrate as \(\int \frac{1}{4} \, du\) resulting in \(\frac{1}{4}(1+x^4) + C_2\).

Integration returns two expressions which can then be set equal to one another to solve for the function \(y\). Integrating is crucial to forming the relationship you need to solve the differential equation.

Solving Differential Equations

Solving differential equations involves finding a function or set of functions that satisfy the original equation. Once both sides have been integrated, the next step is to combine the results and solve for the dependent variable, if possible, to find an explicit solution.
For our case, after integrating:

• \(\frac{1}{2}y^2 + C_1 = \frac{1}{4}(1 + x^4) + C_2\)

Combine constants \(C_1\) and \(C_2\) into a new constant \(C\). The equation simplifies to:

• \(\frac{1}{2}y^2 = \frac{1}{4}(1 + x^4) + C\)

By solving for \(y\), you can express the family of solutions explicitly as \(y = \sqrt{\frac{1 + x^4}{2} + C}\). This process transforms a differential equation into an explicit solution, providing a function \(y\) in terms of \(x\), completing the problem solving process for differential equations using separation of variables.