Exercise Questions

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

Question Statement Summary	Link
Solve separable ODE with exponential term $e^{- 3 x}$	[4.2 Q-1→]
Solve separable ODE with linear $x$ and $y$ terms	[4.2 Q-2→]
Solve separable ODE with power functions $y^{3}$ and $x^{- 2}$	[4.2 Q-3→]
Solve separable ODE using exponent laws for $e^{2 x + 3 y}$	[4.2 Q-4→]
Solve separable ODE with rational function of $x$	[4.2 Q-5→]
Solve separable ODE by rearranging differentials $d y$ and $d x$	[4.2 Q-6→]
Solve separable ODE involving $y^{2}$ and $sin x$	[4.2 Q-7→]
Solve separable ODE with mixed trigonometric functions	[4.2 Q-8→]
Solve IVP for $\frac{d y}{d x} = cos x$ with $y (0) = 1$	[4.2 Q-9→]
Solve IVP for $2 \frac{d y}{d x} = 4 x e^{- x}$ with $y (0) = 2$	[4.2 Q-10→]
Solve IVP with rational coefficient $\frac{1 + x}{x}$ and $y (1) = 1$	[4.2 Q-11→]
Solve IVP involving $tan 2 x$ with $y (0) = 2$	[4.2 Q-12→]
Solve IVP resulting in an inverse tangent form with $y (0) = - 2$	[4.2 Q-13→]
Solve IVP with reciprocal $y$ term and $y (0) = 2$	[4.2 Q-14→]
Solve IVP with polynomial terms and $y (0) = - 1$	[4.2 Q-15→]

Key Concepts

This exercise focuses on the following concepts:

Separation of Variables: The primary technique for solving first-order differential equations by grouping all $y$ terms with $d y$ and all $x$ terms with $d x$ .
First-Order Ordinary Differential Equations (ODEs): Equations involving the first derivative of a function.
General vs. Particular Solutions: Finding the family of solutions (including constant $C$ ) versus finding a specific solution using initial conditions.
Initial Value Problems (IVP): Using a given point $(x_{0}, y_{0})$ to solve for the constant of integration.
Integration Techniques: Application of power rules, exponential integration, trigonometric integration, and integration by parts.

Important Formulas

Below are the key formulas used in this exercise:

Concept	Formula
Separable Form	$\frac{d y}{d x} = f (x) g (y) ⟹ \frac{1}{g ( y )} d y = f (x) d x$
General Solution	$\int \frac{1}{g ( y )} d y = \int f (x) d x + C$
Exponent Laws	$e^{A + B} = e^{A} \cdot e^{B}$
Trig Identity	$cot y = \frac{c o s y}{s i n y}$

Summary

This exercise covers the fundamental method of Separation of Variables for solving first-order differential equations. The general strategy involves algebraic manipulation to isolate variables, followed by integration of both sides. A significant portion of the exercise focuses on Initial Value Problems (IVPs), where students must first find the general solution and then substitute the given initial values to determine the specific value of the integration constant $C$ . Key challenges include identifying the correct integration technique (such as $u$ -substitution or integration by parts) once the variables are separated.

Exercise Questions

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

Question Statement Summary	Link
Solve separable ODE with exponential term $e^{- 3 x}$	[4.2 Q-1→]
Solve separable ODE with linear $x$ and $y$ terms	[4.2 Q-2→]
Solve separable ODE with power functions $y^{3}$ and $x^{- 2}$	[4.2 Q-3→]
Solve separable ODE using exponent laws for $e^{2 x + 3 y}$	[4.2 Q-4→]
Solve separable ODE with rational function of $x$	[4.2 Q-5→]
Solve separable ODE by rearranging differentials $d y$ and $d x$	[4.2 Q-6→]
Solve separable ODE involving $y^{2}$ and $sin x$	[4.2 Q-7→]
Solve separable ODE with mixed trigonometric functions	[4.2 Q-8→]
Solve IVP for $\frac{d y}{d x} = cos x$ with $y (0) = 1$	[4.2 Q-9→]
Solve IVP for $2 \frac{d y}{d x} = 4 x e^{- x}$ with $y (0) = 2$	[4.2 Q-10→]
Solve IVP with rational coefficient $\frac{1 + x}{x}$ and $y (1) = 1$	[4.2 Q-11→]
Solve IVP involving $tan 2 x$ with $y (0) = 2$	[4.2 Q-12→]
Solve IVP resulting in an inverse tangent form with $y (0) = - 2$	[4.2 Q-13→]
Solve IVP with reciprocal $y$ term and $y (0) = 2$	[4.2 Q-14→]
Solve IVP with polynomial terms and $y (0) = - 1$	[4.2 Q-15→]

Key Concepts

This exercise focuses on the following concepts:

Separation of Variables: The primary technique for solving first-order differential equations by grouping all

y

terms with

d y

and all

x

terms with

d x

First-Order Ordinary Differential Equations (ODEs): Equations involving the first derivative of a function.

General vs. Particular Solutions: Finding the family of solutions (including constant

C

) versus finding a specific solution using initial conditions.

Initial Value Problems (IVP): Using a given point

(x_{0}, y_{0})

to solve for the constant of integration.

Integration Techniques: Application of power rules, exponential integration, trigonometric integration, and integration by parts.

Concept

Formula

Separable Form

\frac{d y}{d x} = f (x) g (y) ⟹ \frac{1}{g ( y )} d y = f (x) d x

General Solution

\int \frac{1}{g ( y )} d y = \int f (x) d x + C

Exponent Laws

e^{A + B} = e^{A} \cdot e^{B}

Trig Identity

cot y = \frac{c o s y}{s i n y}

Summary

C

. Key challenges include identifying the correct integration technique (such as

u

-substitution or integration by parts) once the variables are separated.

4.2 Exercise 4.2

Exercise Questions

Key Concepts

Important Formulas

Summary

4.2 Exercise 4.2

Exercise Questions

Key Concepts

Important Formulas

Summary