Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Question Statement Summary	Link
Find order and degree of differential equations	[4.1 Q-1→]
Eliminate arbitrary constants from equations	[4.1 Q-2→]
Verify function as a solution to a differential equation	[4.1 Q-3→]
Find order and degree of $d y - sin x d x = 0$	[4.1 Q-4→]
Verify $y = a cos x + b sin x$ as a solution	[4.1 Q-5→]
Show $y_{1} = x^{2}$ and $y_{2} = x^{3}$ are solutions to a differential equation	[4.1 Q-6→]

Key Concepts

This exercise focuses on the following concepts:

Order of a Differential Equation: The highest order derivative present in the equation.
Degree of a Differential Equation: The power of the highest order derivative, provided the equation is a polynomial in its derivatives.
Elimination of Arbitrary Constants: The process of forming a differential equation by differentiating a general solution and substituting constants.
Verification of Solutions: Checking if a given function satisfies a differential equation by calculating its derivatives and substituting them back into the equation.

Important Formulas

Below are the key formulas used in this exercise:

Concept	Definition / Formula
Order	Highest $n$ in $\frac{d ^{n} y}{d x ^{n}}$
Degree	Power $k$ in $(\frac{d ^{n} y}{d x ^{n}})^{k}$ (after removing radicals)
Solution Verification	Substitute $y, y^{'}, y^{''}, \dots$ into $F (x, y, y^{'}, \dots) = 0$

Summary

This exercise introduces the foundational properties of differential equations. Key learnings include identifying the order (highest derivative) and degree (power of the highest derivative) of an equation. It also covers the relationship between general solutions and differential equations, specifically how to eliminate arbitrary constants through differentiation and how to verify if a specific function is a valid solution to a given differential equation.

Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Question Statement Summary	Link
Find order and degree of differential equations	[4.1 Q-1→]
Eliminate arbitrary constants from equations	[4.1 Q-2→]
Verify function as a solution to a differential equation	[4.1 Q-3→]
Find order and degree of $d y - sin x d x = 0$	[4.1 Q-4→]
Verify $y = a cos x + b sin x$ as a solution	[4.1 Q-5→]
Show $y_{1} = x^{2}$ and $y_{2} = x^{3}$ are solutions to a differential equation	[4.1 Q-6→]

Key Concepts

This exercise focuses on the following concepts:

Order of a Differential Equation: The highest order derivative present in the equation.

Degree of a Differential Equation: The power of the highest order derivative, provided the equation is a polynomial in its derivatives.

Elimination of Arbitrary Constants: The process of forming a differential equation by differentiating a general solution and substituting constants.

Verification of Solutions: Checking if a given function satisfies a differential equation by calculating its derivatives and substituting them back into the equation.

Concept

Definition / Formula

Order

Highest

n

\frac{d ^{n} y}{d x ^{n}}

Degree

Power

k

(\frac{d ^{n} y}{d x ^{n}})^{k}

(after removing radicals)

Solution Verification

Substitute

y, y^{'}, y^{''}, \dots

into

F (x, y, y^{'}, \dots) = 0

Summary

4.1 Exercise 4.1

Exercise Questions

Key Concepts

Important Formulas

Summary

4.1 Exercise 4.1

Exercise Questions

Key Concepts

Important Formulas

Summary