Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 4
\(\frac{d y}{d x}+\frac{3 y}{x}=3 x^{2} ; \quad y=\frac{x^{3}}{2}+\frac{c}{x^{3}}\)
Short Answer
Expert verified
The given function solves the differential equation.
Step by step solution
01
Check Given Differential Equation Solution
To verify whether the given function \(y = \frac{x^3}{2} + \frac{c}{x^3}\) solves the differential equation, we need to find the derivative \(\frac{dy}{dx}\) and substitute back into the differential equation.
02
Differentiate the Given Function
The function is \(y = \frac{x^3}{2} + \frac{c}{x^3}\). Differentiate each term with respect to \(x\): - The derivative of \(\frac{x^3}{2}\) is \(\frac{3x^2}{2}\).- The derivative of \(\frac{c}{x^3}\) is \(-3\frac{c}{x^4}\).So, \(\frac{dy}{dx} = \frac{3x^2}{2} - \frac{3c}{x^4}\).
03
Substitute into the Differential Equation
Substitute \(\frac{dy}{dx}\) and \(y\) into the left-hand side of the differential equation \(\frac{d y}{d x} + \frac{3 y}{x}\):\[\frac{3x^2}{2} - \frac{3c}{x^4} + \frac{3}{x}\left(\frac{x^3}{2} + \frac{c}{x^3}\right)\].
04
Simplify the Expression
Simplify \[ \frac{3x^2}{2} - \frac{3c}{x^4} + \frac{3x^3}{2x} + \frac{3c}{x^4} \]. The terms \(-\frac{3c}{x^4}\) and \(+\frac{3c}{x^4}\) cancel each other. Combine the remaining terms:\[\frac{3x^2}{2} + \frac{3x^2}{2} = 3x^2 \].
05
Verify the Solution
We have simplified the left-hand side to \(3x^2\), which matches the right-hand side of the original differential equation. This verifies that the given function solves the differential equation.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solution Verification
Verifying solutions to differential equations is an essential step to ensure that a supposed solution actually satisfies the equation. This process often involves substituting a proposed function into the original equation and then simplifying to check if both sides are equal.
- Start with the differential equation, such as \( \frac{d y}{d x} + \frac{3 y}{x} = 3 x^{2} \).
- Identify the solution to verify. Here, the provided solution is \( y = \frac{x^3}{2} + \frac{c}{x^3} \).
- Using the derived equation, compute the derivative, substitute it back in, and compare both sides of the original equation.
- If the outcome on both sides matches, your solution is verified and correct.
Derivative Calculation
Calculating derivatives is a fundamental skill in solving differential equations. It allows you to express how a function changes, which is crucial for understanding and solving differential equations.For the given function \( y = \frac{x^3}{2} + \frac{c}{x^3} \):
- Apply the power rule to each term separately.
- The derivative of \( \frac{x^3}{2} \) with respect to \( x \) is \( \frac{3x^2}{2} \).
- The derivative of \( \frac{c}{x^3} \) is found using the rule for negative exponents, resulting in \(-3\frac{c}{x^4} \).
Substitution Method
The substitution method is employed to check if a function is a solution to a differential equation. It involves inserting the proposed function and its derivative into the equation.Steps for substitution:
- Begin by substituting the derivative and the original function into the differential equation.
- For instance, insert \( \frac{dy}{dx} = \frac{3x^2}{2} - \frac{3c}{x^4} \) and \( y = \frac{x^3}{2} + \frac{c}{x^3} \) into the equation \( \frac{d y}{d x} + \frac{3 y}{x} \).
- Simplify the resulting expression: eliminate terms like \(-\frac{3c}{x^4} + \frac{3c}{x^4} \), and combine like terms.
