Question

Determining a Solution In Exercises \(21-28\) , determine whether the function is a solution of the differential equation \(x y^{\prime}-2 y=x^{3} e^{x} .\) $$ y=x^{2} e^{x}-5 x^{2} $$

Short answer

Yes, the function \(y=x^{2} e^{x}-5 x^{2}\) is a solution to the given differential equation \(x y^{\prime}-2 y=x^{3} e^{x}\).

Step-by-step solution

Compute the derivative of the function

First, let's compute the derivative of the function \(y\). The derivative of \(y=x^{2} e^{x}-5 x^{2}\) can be done using the product and power rules. The derivative \(y'\) is \(y'=2x e^{x}+x^{2} e^{x}-10x\).

Substitute y and y' into the differential equation

Next, we substitute \(y\) and \(y'\) into the differential equation \(x y^{\prime}-2 y=x^{3} e^{x}\). By doing this, we obtain:\(x(2x e^{x}+x^{2} e^{x}-10x)-2(x^{2} e^{x}-5 x^{2})\). Simplifying this expression leads us to \(2x^{2} e^{x}+x^{3} e^{x}-10x^{2}-2x^{2} e^{x}+10x^{2}\).

Check if the resulting expression equals to \(x^{3} e^{x}\)

Finally, we have to determine whether the resulting expression from step 2 equals to \(x^{3} e^{x}\). After simplifying the expression from step 2, we are left with \(x^{3} e^{x}\), which is indeed the right-hand side of the differential equation.

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. A derivative represents an instantaneous rate of change, essentially telling us how a function's output value changes as its input value changes. In the context of our exercise, the differentiation process is used to find the derivative of the function y = x^2 e^x - 5x^2. Understanding differentiation is crucial because in differential equations, we often deal with relationships between functions and their derivatives.

For students facing challenges with differentiation, it's essential to grasp that taking the derivative is like uncovering the underlying rate at which one variable changes concerning another. Mastery in this area is achieved through practice and familiarity with derivative rules, like the power and product rules, which leads us to our next crucial concept.

Product Rule

The product rule is a derivative rule used when differentiating products of two or more functions. It states that the derivative of a product of functions is the first function times the derivative of the second function, plus the second function times the derivative of the first function. Formally, if we have two functions u(x) and v(x), then the derivative of their product u(x)v(x) is u'(x)v(x) + u(x)v'(x).

In our exercise, we apply the product rule to differentiate x^2e^x, resulting in 2xe^x + x^2e^x, after also applying the power rule to the x^2 term. It is a common pitfall for students to forget to apply the product rule whenever they encounter a product of functions and default to simpler rules. Clear understanding and practice of the product rule are pivotal in solving differential equations that involve products of functions.

Substitution in Differential Equations

Substitution in differential equations is a technique used to simplify equations and facilitate their solving process. It involves replacing one part of the equation with another equivalent part, often making it easier to integrate or differentiate. In the case of our exercise, substitution comes into play after we've found the derivatives. We then substitute the original function y and its derivative y' back into the given differential equation, xy' - 2y = x^3e^x.

One common error during substitution is misalignment of variables or forgetting to substitute all occurrences of the variables. It is imperative for learners to pay careful attention to every term and ensure that all necessary substitutions are made before proceeding with further simplifications.

Verifying Solutions to Differential Equations

After solving a differential equation or being given a potential solution, it's crucial to verify that the solution is indeed correct. Verification involves substituting the solution into the original differential equation and checking if the equation holds true. For our exercise, the final step entails ensuring the simplified expression after substitution matches the right-hand side of the differential equation, in this case, x^3e^x.

If both sides of the differential equation are identical upon substitution, then the proposed function is indeed a solution to the differential equation. This verification not only checks the solution's validity but also reinforces the understanding of the relationship between functions and derivatives, emphasizing the critical thinking aspect of differential equations. Students should always make it a practice to verify solutions, as it's an excellent habit that helps avoid costly mistakes in both homework and exams.