Question

Solve the given Bernoulli equation. $$ x^{2} y^{\prime}+2 y=2 e^{1 / x} y^{1 / 2} $$

Short answer

Question: Solve the Bernoulli equation \(x^2y^\prime + 2y = 2e^{1/x}y^{1/2}\) and provide the implicit solution for y. Answer: The implicit solution for y is given by \(y = \left(e^{-x^2} \left(\int 2x^2 e^{x^2 + 1/x} dx + C\right)\right)^2\).

Step-by-step solution

Divide through by \(y^n\)

For the given equation, we have: $$ x^2y^\prime + 2y = 2e^{1/x}y^{1/2} $$ We can see that in this case, \(n = 1/2\). So, we will divide through by \(y^{1/2}\): $$ x^2 \frac{y^\prime}{y^{1/2}} + 2 \frac{y}{y^{1/2}} = 2e^{1/x} $$ This simplifies to: $$ x^2 \frac{y^\prime}{{y^\prime}^{1/2}} + 2{y^{1/2}} = 2e^{1/x} $$

Apply the substitution \(u = y^{1-n}\)

In this case, we have \(n = 1/2\), so \(1 - n = 1 - 1/2 = 1/2\). Hence, we make the substitution \(u = y^{1/2}\), and its derivative is \(u^\prime = \frac{1}{2}y^{-1/2}y^\prime\). From the substitution, we have \(u = {y^{1/2}}\), so we can rewrite the equation in terms of \(u\): $$ x^2 \frac{2u^\prime}{u} + 2u = 2e^{1/x} $$ This simplifies to: $$ 2x^2u^\prime + 2x^2u = 2x^2e^{1/x} $$

Solve the resulting linear first-order differential equation

Now, the equation is in the form of a linear first-order differential equation: $$ 2x^2u^\prime + 2x^2u = 2x^2e^{1/x} $$ To solve this, we first find the integrating factor, which is given by \(e^{\int 2x\,dx} = e^{x^2}\). Then, we multiply the equation by the integrating factor and integrate: $$ e^{x^2}(2x^2u^\prime + 2x^2u) = 2x^2e^{x^2 + 1/x} $$ $$ \frac{d}{dx}(u e^{x^2}) = 2x^2e^{x^2 + 1/x} $$ Now, integrate both sides: $$ u e^{x^2} = \int 2x^2 e^{x^2 + 1/x} dx + C $$ Then, isolate \(u\): $$ u = e^{-x^2} \left(\int 2x^2 e^{x^2 + 1/x} dx + C\right) $$

Re-introduce \(y\) and solve for \(y\)

Now, recall the substitution we made earlier: \(u = y^{1/2}\). Therefore, to find \(y\), we will first express \(y\) in terms of \(u\): $$ y = u^2 $$ Now, substitute \(u\) back into the equation: $$ y = \left(e^{-x^2} \left(\int 2x^2 e^{x^2 + 1/x} dx + C\right)\right)^2 $$ This is the implicit solution for the given Bernoulli equation.

Key concepts

Differential Equations

Differential equations are equations that relate a function to its derivatives. They are essential in describing various phenomena in engineering, physics, economics, and biology. These equations can model change and predict future behavior based on current conditions. There are many types of differential equations, but they all share a common feature: they contain unknown functions and their derivatives.
A solution to a differential equation is a function that satisfies the equation when substituted into it.
Depending on the complexity, solutions can be explicit formulas, implicit equations, or even numerical approximations. In solving differential equations, methods vary widely and often depend on the specific form of the equation.

Linear First-Order Differential Equation

A linear first-order differential equation is one of the simplest types of differential equations. It takes the standard form \( dy/dx + P(x)y = Q(x) \), where \( P(x) \) and \( Q(x) \) are functions of \( x \). Such equations are significant because they can be solved directly using straightforward methods.
The simplicity of linear first-order differential equations means they frequently model real-world processes where changes are proportional to the current value, such as cooling or heating processes, population growth, or chemical reactions.

• The main task in solving these equations is to find an appropriate integrating factor to simplify the problem.
• This process typically involves multiplying the entire equation by a specific function to make it easier to solve.

Understanding this approach provides a foundation for tackling more complex differential equations later on.

Integrating Factor

The integrating factor is a crucial tool when solving linear first-order differential equations. It is a function, usually denoted as \( \mu(x) \), that simplifies the differential equation to a form that can be directly integrated.
To find this factor, you need to calculate: \[ \mu(x) = e^{\int P(x) \, dx} \] In the context of our problem, the integrating factor is used to multiply throughout the transformed linear equation.
This process converts the equation into an exact differential, allowing for straightforward integration.

• The integrating factor essentially "unravels" the complexity of the equation, helping to construct the solution path.
• After applying the integrating factor, the left side of the differential equation typically becomes the derivative of a product, which is much easier to manage.

Once integrated, the solution generally includes an arbitrary constant of integration, which can be defined based on initial conditions if provided.

Mathematical Substitution

Mathematical substitution is a powerful technique used to simplify and solve differential equations, especially those that are not linear initially. By substituting variables with more manageable expressions, complex equations become more solvable.
For the Bernoulli equation, a specific substitution \( u = y^{1-n} \) is used to reduce the problem to a linear first-order differential equation.

• This substitution transforms the original nonlinear equation into a linear form, which can subsequently be solved using methods like the integrating factor.
• Substitution often involves derivatives, requiring us to express derivatives in terms of the new variables as well.

After solving the simplified equation, the final step is to substitute back to the original variables, obtaining the explicit or implicit solution to the initial problem. Each successful substitution targets making the differential equation as manageable as possible.