Question

Solving a Bernoulli Differential Equation In Exercises \(59-66,\) solve the Bernoulli differential equation. The Berchoulli equation is a well-known nonlinear equation of the form $$ y^{\prime}+P(x) y=Q(x) y^{n} $$ that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is $$ y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{f(1-n) P(x) d x} d x+C $$ $$ y y^{\prime}-2 y^{2}=e^{x} $$

Short answer

The detailed steps and calculations following the steps would ultimately lead us to the solution for \(y(x)\). This is a comprehensive problem and involves detailed steps, hence the exact short answer is not available unless the integration in Step 3 is performed.

Step-by-step solution

Apply the substitution

We begin by making the required substitution to reduce the Bernoulli equation into a linear form. Looking at the general solution for a Bernoulli equation, the substitution \(z = y^{1-n}\) can make our equation linear, where \(n = 2\). Hence our \(z\) will be \(z = y^{1-2} = y^{-1}\). Differentiate \(z = y^{-1}\) with respect to \(x\), we get \(z' = -y^{-2}y'\).

Replace \(y\) and \(y'\) in terms of \(z\) in the original equation

By replacing \(yy'-2y^2\) by \(z\) and \(z'\) respectively, the original Bernoulli equation \(yy'-2y^2 = e^x\) is changed to its equivalent linear form. First, we replace \(yy'\) by \(-z^2z'\) and \(y^2\) by \(z^{-2}\). So our equation becomes \(-z^2z' - 2z^{-2} = e^x\). Multiply the whole equation by \(-z^{-2}\) to give us a standard first order linear equation. The equation becomes \(z' + 2z = -z^2e^x\).

Solve the resulting first order linear differential equation

We now have a first-order linear differential equation of the form \(\frac{dz}{dx}+p(x)z=q(x)\) where \(p(x) = 2\) and \(q(x) = -z^2e^x\) which can be solved through the method of integrating factors. However, this equation is a separable equation and can be separated into variables. Rewrite our equation as \(\frac{dz}{dx} + z^2e^x = 2z\) and separate the variables: \(\frac{dz}{z^2e^x - 2z} = dx\). Next, perform the integration: \(\int\frac{dz}{z^2e^x - 2z} = \int dx\). After you perform the integration you obtain \(z\) as a function of \(x\). We need to revert back to \(y\) terms for our final solution. To do this, replace \(z\) by \(1/y\) to get our \(y\) as a function of \(x\).

Replace \(z\) with \(y^{-1}\)

Finally, we replace \(z\) by \(y^{-1}\) in the solution from Step 3, and find \(y\) as a function of \(x\). Solving for \(y\), gives us our final solution.

Key concepts

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. In essence, it describes how a particular quantity changes over time or space, and such equations are foundational in expressing the laws of physics and other disciplines. Differential equations can be simple or complex, linear or nonlinear, and they dictate the rate at which relationships vary.

To solve a differential equation means to find a function or set of functions that satisfy the equation. Examples include the Newton's Law of Cooling, the decay of radioactive elements, and population models. The Bernoulli differential equation we are examining here is a specific type of nonlinear differential equation that can be reduced to a linear form with an appropriate substitution.

Integrating Factors

An integrating factor is a mathematical tool used to solve first-order linear differential equations. The technique involves multiplying the whole equation by an integrating factor, which is cleverly selected to convert the equation into an exact differential. This exact differential can then be integrated to find the solution.

Creating an Exact Differential

The beauty of an integrating factor is that it transforms a product into a derivative of a product, which makes it possible to bring the equation into a form that can be integrated directly. In our Bernoulli equation example, we can use an integrating factor to solve the equivalent linear form of the original nonlinear equation.

Separable Equations

Separable equations are a class of differential equations in which the variables can be separated on opposite sides of the equation. In other words, all terms with the dependent variable can be moved to one side of the equation, and all terms with the independent variable can be moved to the other.

By performing this separation, each side of the equation becomes solely a function of one variable, allowing us to integrate both sides independently. This is a commonly used technique for solving many first-order differential equations, just like we see in Step 3 of our problem-solving process for the Bernoulli equation.

Substitution Method

The substitution method involves replacing variables or expressions within an equation with another set of variables or expressions to simplify the problem. It's a powerful approach for transforming complicated equations into simpler ones that are more manageable to solve.

For the Bernoulli differential equation in our example, we used the substitution method to convert a nonlinear equation into a linear one by setting a new variable equal to a power of the original function, simplifying the process to find the general solution. This technique demonstrates how we can handle more complex problems by reducing them to forms that we have the tools to solve.