Basics of Differential Equations

A differential equation is an equation involving derivatives of a function with respect to one or more independent variables. The derivative of the function is denoted by \( \frac{dy}{dx} \).

Order of a differential equation is the order of the highest derivative present in the equation.

Degree of a differential equation is the power of the highest order derivative after removing radicals and fractions.

Notation for derivatives:

• \( \frac{dy}{dx} = y' \)
• \( \frac{d^2y}{dx^2} = y'' \)
• \( \frac{d^3y}{dx^3} = y''' \)
• For higher order derivatives, \( y^{(n)} = \frac{d^n y}{dx^n} \)

Order and degree are always positive integers.

Example 1

Find the order and degree of the differential equation \( y^{(3)} + y^2 + e^{y'} = 0 \).

Solution: The highest order derivative is \( y^{(3)} \), so the order is 3. The equation is not a polynomial in derivatives due to the exponential term \( e^{y'} \), so the degree is not defined.

Practice Set

• Find the order and degree of \( \frac{d^2y}{dx^2} + 3y' + y = 0 \).
• Determine the order and degree of \( (y'')^2 + y = 5 \).
• Find the order and degree of \( \sqrt{y''} + y = 0 \).

Answer Key

• Order = 2, Degree = 1
• Order = 2, Degree = 2
• Order = 2, Degree not defined (due to square root)

Quick Reference

Term	Definition
Order	Highest order derivative in the equation
Degree	Power of highest order derivative after simplification

Glossary

• Derivative: Rate of change of a function.
• Order: Highest derivative order in the equation.
• Degree: Power of the highest order derivative.

Variable Separable Methods

General Solution: Solution containing arbitrary constants equal to the order of the differential equation.

Particular Solution: Solution obtained by assigning specific values to arbitrary constants in the general solution.

A differential equation is separable if it can be written as \( f(x) dx = g(y) dy \). The solution is obtained by integrating both sides:

\[ \int f(x) dx = \int g(y) dy + k \]

where \( k \) is the constant of integration.

Example 2

Find the general solution of \( \frac{dy}{dx} = x \).

Solution:

Separate variables:

\( dy = x dx \)

Integrate both sides:

\[ \int dy = \int x dx \]

\[ y = \frac{x^2}{2} + c \]

Practice Set

• Solve \( \frac{dy}{dx} = 3x^2 \).
• Solve \( \frac{dy}{dx} = \frac{y}{x} \).
• Solve \( \frac{dy}{dx} = \frac{x^2}{y^2} \).

Answer Key

• \( y = x^3 + c \)
• Separate variables: \( \frac{dy}{y} = \frac{dx}{x} \), integrate to get \( \ln|y| = \ln|x| + c \), so \( y = kx \).
• Separate variables: \( y^2 dy = x^2 dx \), integrate to get \( \frac{y^3}{3} = \frac{x^3}{3} + c \), so \( y^3 = x^3 + k \).

Quick Reference

Step	Action
1	Rewrite as \( f(x) dx = g(y) dy \)
2	Integrate both sides
3	Add constant of integration

Glossary

• General Solution: Solution with arbitrary constants.
• Particular Solution: Solution with constants assigned specific values.
• Separable Equation: Equation that can be written as product of functions of x and y separately.

Linear Differential Equations

A first order linear differential equation in \( y \) is of the form:

\[ \frac{dy}{dx} + P(x) y = Q(x) \]

where \( P(x) \) and \( Q(x) \) are functions of \( x \) only.

Solution method:

1. Find the integrating factor (I.F.):

\[ I.F. = e^{\int P(x) dx} \]

2. Multiply the entire differential equation by the I.F.:

\[ I.F. \cdot \frac{dy}{dx} + I.F. \cdot P(x) y = I.F. \cdot Q(x) \]

3. Left side becomes derivative of \( y \cdot I.F. \):

\[ \frac{d}{dx} \left( y \cdot I.F. \right) = I.F. \cdot Q(x) \]

4. Integrate both sides:

\[ y \cdot I.F. = \int I.F. \cdot Q(x) dx + c \]

5. Finally, solve for \( y \):

\[ y = \frac{1}{I.F.} \left( \int I.F. \cdot Q(x) dx + c \right) \]

Similarly, a linear differential equation in \( x \) is of the form:

\[ \frac{dx}{dy} + P(y) x = Q(y) \]

and solved by analogous steps using integrating factor \( e^{\int P(y) dy} \).

Example 3

Solve \( \frac{dy}{dx} + y = x \).

Solution:

Given \( P(x) = 1 \), \( Q(x) = x \).

Integrating factor:

\[ I.F. = e^{\int 1 dx} = e^x \]

Multiply both sides by \( e^x \):

\[ e^x \frac{dy}{dx} + e^x y = x e^x \]

Left side is derivative of \( y e^x \):

\[ \frac{d}{dx} (y e^x) = x e^x \]

Integrate both sides:

\[ y e^x = \int x e^x dx + c \]

Use integration by parts for \( \int x e^x dx \):

Let \( u = x \), \( dv = e^x dx \), then \( du = dx \), \( v = e^x \).

\[ \int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + c_1 \]

Therefore,

\[ y e^x = x e^x - e^x + c \]

Divide both sides by \( e^x \):

\[ y = x - 1 + c e^{-x} \]

Practice Set

• Solve \( \frac{dy}{dx} + 2y = e^{-x} \).
• Solve \( \frac{dy}{dx} - y = x \).
• Solve \( \frac{dx}{dy} + 3x = y^2 \).

Answer Key

• Integrating factor \( e^{2x} \), solution \( y = e^{-2x} \int e^{x} dx + c e^{-2x} = \frac{1}{3} e^{-x} + c e^{-2x} \).
• Integrating factor \( e^{-x} \), solution \( y = e^{x} \int x e^{-x} dx + c e^{x} = -x -1 + c e^{x} \).
• Integrating factor \( e^{3y} \), solution \( x e^{3y} = \int y^2 e^{3y} dy + c \) (integration by parts required).

Quick Reference

Step	Action
1	Write in standard form \( \frac{dy}{dx} + P(x) y = Q(x) \)
2	Calculate integrating factor \( e^{\int P(x) dx} \)
3	Multiply equation by integrating factor
4	Integrate both sides
5	Solve for \( y \)

Glossary

• Integrating Factor: Function used to simplify linear differential equations.
• Linear Differential Equation: Equation linear in the dependent variable and its derivatives.

Homogeneous Differential Equations

A first order differential equation is homogeneous if it can be written as:

\[ \frac{dy}{dx} = f(x,y) \]

where \( f(kx, ky) = k^n f(x,y) \) for some degree \( n \).

Method of solution:

Substitute \( y = v x \) (where \( v \) is a function of \( x \)) so that:

\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]

Rewrite the differential equation in terms of \( v \) and \( x \), then separate variables and integrate.

Example 4

Solve \( y' = \frac{x + y}{x} \).

Solution:

Rewrite:

\[ \frac{dy}{dx} = 1 + \frac{y}{x} \]

Substitute \( y = v x \), so \( \frac{dy}{dx} = v + x \frac{dv}{dx} \).

Substitute into equation:

\[ v + x \frac{dv}{dx} = 1 + v \]

Rearranged:

\[ x \frac{dv}{dx} = 1 \]

Separate variables:

\[ \frac{dv}{dx} = \frac{1}{x} \implies dv = \frac{dx}{x} \]

Integrate both sides:

\[ \int dv = \int \frac{dx}{x} \implies v = \ln |x| + c \]

Recall \( v = \frac{y}{x} \), so:

\[ \frac{y}{x} = \ln |x| + c \implies y = x \ln |x| + c x \]

Practice Set

• Solve \( \frac{dy}{dx} = \frac{y}{x} + \frac{x}{y} \).
• Solve \( \frac{dy}{dx} = \frac{x^2 + y^2}{x y} \).
• Solve \( \frac{dx}{dy} = \frac{x + y}{y} \).

Answer Key

• Substitute \( y = v x \), solve to get \( y = c x \).
• Rewrite and substitute, integrate to find implicit solution involving logarithms.
• Substitute \( x = v y \), solve for \( v \) and integrate accordingly.

Quick Reference

Step	Action
1	Check if \( f(kx, ky) = k^n f(x,y) \)
2	Substitute \( y = v x \) or \( x = v y \)
3	Rewrite \( \frac{dy}{dx} \) or \( \frac{dx}{dy} \) in terms of \( v \) and \( x \) or \( y \)
4	Separate variables and integrate

Glossary

• Homogeneous Function: Function satisfying \( f(kx, ky) = k^n f(x,y) \).
• Substitution: Replacing variables to simplify the equation.