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 $$ y^{1-n} e^{\int(1-n) P(x) d x}=\int(1-n) Q(x) e^{f(1-n) P(x) d x} d x+C $$ $$ y^{\prime}-y=e^{x \sqrt[3]{y}} $$

Short answer

The solution to the Bernoulli equation is \(y = xe^x + Ce^x\).

Step-by-step solution

Identifying Coefficients

Identify values for P(x), Q(x) and n by rewriting equation \(y'-y=e^{x\sqrt[3]{y}}\) this becomes \((y'-y)-(e^{x\sqrt[3]{y}})=0\) which aligns to the standard form \(y'+P(x)y=Q(x)y^n\). Therefore, P(x)=-1, Q(x) =\(e^{x \sqrt[3]{y}}\) and n=0.

Applying Substitution

Reduce the Bernoulli equation to a linear equation. Use substitution \(v=y^{1-n} = y^1 = y\). The derivative of v is \(v'=dv/dy=1*y'=y'\). Substitute v into the equation. Our equation now reads as \(v'+P(x)v=Q(x)\), or substituting in known values, \(v'+(-1)v=e^{x \sqrt[3]{y}}\).

Integrating

Now we solve the linear differential equation. We take the equation to be in this form \(v' + P(x)v = Q(x)\). After substituting the values of P(x) and Q(x) we have, \(v' - v = e^{x}\). We can solve this by finding an integrating factor \(e^{\int P(x) dx} = e^{-x}\). Then multiply every term by this factor: \(e^{-x}v' - e^{-x}v = 1\).

Solving Linear Differential Equation

We can identify that the left-hand side is the derivative of \(e^{-x}v\) with respect to x. So we write the equation as \((e^{-x}v)' = 1\). Integrating both sides with respect to x gives \(e^{-x}v = x + C\). So, v = y = \(xe^x + Ce^x\) (multiplying through by \(e^x\)).

Retrieve the original variable y

Finally, recall the substitution we made at step 2. We need to express v (that is currently equal to y) as a function of y. \(y = xe^x + Ce^x\). Using the original equation, we modify to express y in terms of x: \(y = xe^x + Ce^x\). This is the solution to our problem.

Key concepts

Nonlinear Differential Equation

In the world of differential equations, nonlinear differential equations present a unique challenge. Unlike their linear counterparts, these equations involve variables raised to a power greater than one or involve products of variables, which makes them more complex to solve. A Bernoulli Differential Equation is a classic example of a nonlinear differential equation. It takes the form \[ y' + P(x)y = Q(x)y^n \]and becomes nonlinear due to the term \( y^n \).Understanding nonlinear equations requires familiarity with methods that sometimes transform them into linear ones, which are easier to handle due to their simpler structure. These equations can depict many real-world phenomena where the relationship between variables is not straightforward.

Linear Differential Equation

A linear differential equation is much simpler to handle because it maintains the linearity of variables without involving them in higher powers or products. Once a nonlinear equation, such as a Bernoulli equation, is reduced into linear form, it becomes solvable by traditional methods. For instance, through substitution, we can transform a Bernoulli equation into the standard first-order linear form\[ v' + P(x)v = Q(x) \].Key characteristics of linear differential equations include:

• No exponents greater than one on the derivatives.
• No variable products involving derivatives.
• Can be handled using techniques like integrating factors or separation of variables.

Linear equations play a pivotal role when converting and then solving complex differential equations through methods like integration techniques.

Integration Technique

The integration technique is crucial for solving differential equations, particularly once they have been reduced to a manageable linear form. In our solution process, we utilize an integrating factor to effectively solve the linear equation. This integrating factor is determined as \( e^{\int P(x) \; dx} \).To solve the equation:

• Determine the integrating factor and multiply it through the equation.
• This often rewrites the equation such that the left-hand side becomes a derivative of a product.
• Integrate both sides with respect to the variable.
• Apply boundary or initial conditions to solve for any constants of integration.

This technique simplifies complex differential equations and is foundational in transitioning from results of manipulation back to meaningful solutions.

Substitution Method

The substitution method is a transformative approach used primarily to convert challenging nonlinear equations into simpler linear ones. In our given Bernoulli equation, the substitution \( v = y^{1-n} \)transforms the nonlinear differential equation into a potentially solvable linear form. Once the equation is linearized, conventional methods can be applied to find the solution.Steps commonly followed in the substitution method include:

• Identifying the substitution that simplifies the equation.
• Rewriting the equation using the new variable.
• Solving the resultant equation using techniques like integrating factors.
• Reversing the substitution to express the solution in terms of the original variables.

This powerful method unlocks complex equations, making them tractable and open to analytical solutions.