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Chapter 31: Problem 20

Show that the given equation is a solution of the given differential equation. $$x y^{\prime}-3 y=x^{2}, \quad y=c x^{3}-x^{2}$$

Short Answer

Expert verified
The function \( y = cx^3 - x^2 \) satisfies the differential equation.

Step by step solution

01

Identify the Given Functions and Derivatives

First, identify the functions given in the problem. We have a differential equation \( x y' - 3y = x^2 \) and a potential solution \( y = cx^3 - x^2 \).Differentiate the given function \( y \) with respect to \( x \) to find \( y' \).
02

Differentiate the Given Function

Calculate the derivative of \( y = cx^3 - x^2 \) with respect to \( x \) to find \( y' \).\[ y' = \frac{d}{dx}(cx^3 - x^2) = 3cx^2 - 2x \]
03

Substitute into the Differential Equation

Now substitute \( y = cx^3 - x^2 \) and \( y' = 3cx^2 - 2x \) into the differential equation \( x y' - 3y = x^2 \).This gives us:\[ x(3cx^2 - 2x) - 3(cx^3 - x^2) = x^2 \]
04

Simplify the Equation

Expand the terms:1. Compute \( x(3cx^2 - 2x) = 3cx^3 - 2x^2 \).2. Compute \( 3(cx^3 - x^2) = 3cx^3 - 3x^2 \).Substitute these back into the equation:\[ 3cx^3 - 2x^2 - 3cx^3 + 3x^2 = x^2 \]
05

Combine Like Terms

Combine like terms in the equation:\[ 3cx^3 - 3cx^3 - 2x^2 + 3x^2 = x^2 \]Which simplifies to:\[ x^2 = x^2 \]
06

Verify the Solution

Since both sides of the equation equal \( x^2 \), the original function \( y = cx^3 - x^2 \) is indeed a solution to the differential equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solution Verification
Verifying a solution to a differential equation involves ensuring that the proposed function satisfies the equation. In this case, our goal is to prove that \( y = cx^3 - x^2 \) is a solution to the differential equation \( x y' - 3y = x^2 \). By substituting \( y = cx^3 - x^2 \) and its derivative \( y' = 3cx^2 - 2x \) into the differential equation, we check if the two sides become equal.
  • Substitute \( y \) and \( y' \) into the differential equation.
  • Simplify both sides of the equation.
  • If the left-hand side equals the right-hand side, verification is successful.
In this exercise, substitution results in \( x^2 = x^2 \), showing that the function satisfies the differential equation. Verifying in this way is crucial because it confirms the mathematical validity of the solution.
Differentiation
Differentiation is a fundamental concept in calculus that allows us to determine how a function changes at any given point. In the context of this problem, we need to find the derivative of a potential solution to check if it satisfies the differential equation. The given expression for \( y \) is \( y = cx^3 - x^2 \).
  • Apply differentiation rules: the power rule, where the derivative of \( x^n \) is \( nx^{n-1} \).
  • Calculate the derivative of \( y \): \( y' = 3cx^2 - 2x \).
By differentiating \( y \), we find the rate at which \( y \) changes, which is vital for verifying its role in the initial differential equation. Knowing \( y' \) allows us to correctly substitute back into the equation for verification purposes.
Substitution Method
The substitution method in solving differential equations involves replacing the function and its derivative with their respective expressions in terms of \( x \). This technique is particularly helpful in simplifying and solving equations to check the validity of a potential solution.
  • Substitute \( y = cx^3 - x^2 \) into the differential equation.
  • Replace \( y' \) with its derivative, \( 3cx^2 - 2x \).
  • Simplify the equation using basic algebraic manipulation.
This method is a powerful verification tool. Once substitution is complete, simplifying the equation ensures no errors in calculations and confirms whether the original equation holds true under the proposed solution. This technique provides clarity and a straightforward way to demonstrate solution correctness.
Mathematical Proofs
Mathematical proofs are logical arguments that validate the truth of a mathematical statement. In verifying a solution for a differential equation, the proof involves showing that substituting the function and its derivative results in an identity or equality.
  • Start by writing the differential equation and the proposed solution.
  • Substitute and simplify each side to see if they match.
  • If after simplification both sides are equal, a proof confirming the solution is achieved.
In this exercise, the substitution simplifies to \( x^2 = x^2 \), proving the function \( y = cx^3 - x^2 \) solves the differential equation. Mathematical proofs ensure accuracy and provide confidence in the validity of solutions, which is central to all mathematical disciplines.

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Most popular questions from this chapter

Solve the given differential equations by Laplace transforms. The function is subject to the given conditions. $$2 y^{\prime \prime}+y^{\prime}-y=\sin 3 t, \quad y(0)=0, y^{\prime}(0)=0$$

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