First-order Differential Equations Chapter 2

Overview II. Linear equations Chapter 1 : Introduction to Differential Equations I. Separable variables III. Exact Equations IV. Solution by substitutions

I. Separable variables Learning Objective At the end of this section you should be able to identify and solve a separable DE.

I. Separable variables Definition A first-order DE of the form is said to be separable or to have separable variables. is said to be separable or to have separable variables.

I. Separable variables Examples: is separable Indeed, is not separable Indeed, it cannot be put on the product form

I. Separable variables Remark: By dividing by, the separable DE, can be written in the form: can be written in the form: where

I. Separable variables Method of solution: From the form : we have to integrated both sides. to obtain One-parameter family of implicit or explicit solutions.

I. Separable variables Examples:

I. Separable variables Examples:

I. Separable variables Examples: So, the solution of the IVP:.

I. Separable variables Examples:.

I. Separable variables Examples:. Solution of IVP:

I. Separable variables Exercise I:. Solve the following DE by separation of variables:

II. Linear equations Learning Objective At the end of this section you should be able to identify and solve a linear DE.

II. Linear Equations Definition A first-order DE of the form is said to be linear equation in the dependent variable. is said to be linear equation in the dependent variable.

II. Linear Equations Remark By dividing both sides by, a linear equation can be written in the standard form:

II. Linear Equations Definition Values of that will make, are called singular points of the equation. Example: is a singular point.

II. Linear Equations Definition The function is defined as the integrating factor. Remark:

II. Linear Equations Find the integrating factor for : Examples: 1)

II. Linear Equations Find the integrating factor for : Examples: 2) Singular point :

II. Linear Equations Method of solution 1) Write the standard form : 2) Find the integrating factor :

II. Linear Equations Method of solution 3) Multiply the Standard form by the integrating factor : Standard form :

II. Linear Equations Method of solution 4) Integrate both sides of the last equation :

II. Linear Equations Solve the following linear DE : Examples: 1)

II. Linear Equations Examples: 1)

II. Linear Equations Examples: 2)

II. Linear Equations Examples: 2) Valid on

II. Linear Equations Examples: 3)

II. Linear Equations Exercise-II: Solve the following linear DE :

III. Exact equations Learning Objective At the end of this section you should be able to identify and solve an exact ODE.

III. Exact equations Definition Consider a function of two variables: is the partial derivative of regarding ( is considered as a constant). ( is considered as a constant). is the partial derivative of regarding ( is considered as a constant). ( is considered as a constant).

III. Exact equations Definition Consider a function of two variables: its differential is :

III. Exact equations Example:

III. Exact equations Examples:

III. Exact equations Examples:

III. Exact equations Definition A differential expression : is an exact differential if it corresponds to the differential of some function that means

III. Exact equations Example is an exact differential Indeed

III. Exact equations Definition A DE of the form: is an exact equation if the left side is an exact differential. In that case, the DE is equivalent to An implicit solution will be

III. Exact equations Example is an exact equation Indeed

III. Exact equations Theorem A necessary and sufficient condition that be an exact differential is.

III. Exact equations Example exact equation Indeed

III. Exact equations Method of solution. Step 1: Check exactitude Example: Exact DEthere exists such that ?

III. Exact equations Method of solution. Step 2: integrate regarding constant for but not for

III. Exact equations Method of solution. Step 3: Differentiate regarding

III. Exact equations Method of solution. Step 4: Integrate regarding

III. Exact equations Method of solution. Step 5: Solution An implicit solution is An explicit solution isdefined when

III. Exact equations Example: exact equation Solve the following ODEs:

III. Exact equations Example:

III. Exact equations Example: Implicit solution

III. Exact equations Example: exact equation

III. Exact equations Example:

III. Exact equations Example: Family of implicit solutions Solution of the IVP:

III. Exact equations Exercise-IIIa: Determine whether the given DE is exact. If it is, solve it

III. Exact equations Remark: make exact some non-exact DEs is non exact if where There are cases where the equation can be made exact. How?

III. Exact equations Remark: make exact some non-exact DE compute If the result is a function of the sole variable : then find the integrating factor :

III. Exact equations Remark: make exact some non-exact DE Now Exact! Multiply the DE by the integrating factor :

III. Exact equations Remark: make exact some non-exact DE Example

III. Exact equations Remark: make exact some non-exact DE Example Exact!

III. Exact equations Remark: make exact some non-exact DE compute If is NOT a function of the unique variable then find the integrating factor : If the result is a function of the sole variable :

III. Exact equations Remark: make exact some non-exact DE Example

III. Exact equations Remark: make exact some non-exact DE Example Exact!

III. Exact equations Exercise-IIIb: Solve the given D.E by finding an appropriate integrating factor.

IV. Solution by substitutions Learning Objective At the end of this section you should be able to solve Homogeneous and Bernoulli’s DEs.

IV. Solution by substitutions Bernoulli Equation A DE in the form where is a real number is said a Bernoulli equation. Definition :

IV. Solution by substitutions Bernoulli Equation Example :

IV. Solution by substitutions Bernoulli Equation Substitution: Substitution:

IV. Solution by substitutions Bernoulli Equation

IV. Solution by substitutions Bernoulli Equation Example :

IV. Solution by substitutions Bernoulli Equation Example :

IV. Solution by substitutions Bernoulli Equation Example :

IV. Solution by substitutions Bernoulli Exercise IVb : Solve the given Bernoulli equation by using an appropriate substitution.

IV. Solution by substitutions Homogeneous DE If a function has the property that, for some real number, then is said to be a homogeneous function of degree. Definition :

IV. Solution by substitutions Homogeneous DE is a homogeneous function of degree 2. Example1 :

IV. Solution by substitutions Homogeneous DE is a homogeneous function of degree 2/3. Example 2 :

IV. Solution by substitutions Homogeneous DE is not homogeneous Example 3 : We can’t factorize by a power of

IV. Solution by substitutions Homogeneous DE is homogeneous of degree 0 Example 4 :

IV. Solution by substitutions Homogeneous DE A DE of the form is said to be homogeneous if both and are homogeneous functions of the same degree. Definition :

IV. Solution by substitutions Homogeneous DE are homogeneous of degree 2 Example : is homogeneous. Indeed and

IV. Solution by substitutions Homogeneous DE are homogeneous of degree 1 Method of solution: 1) Homogeneity : and 2) Substitution

IV. Solution by substitutions Homogeneous DE Method of solution:

IV. Solution by substitutions Homogeneous DE Method of solution:

IV. Solution by substitutions Homogeneous DE Method of solution: 2) Substitution (2 nd option)

IV. Solution by substitutions Homogeneous DE Method of solution: 2) Substitution

IV. Solution by substitutions Homogeneous DE Method of solution: 2) Substitution

IV. Solution by substitutions Homogeneous DE Exercise IVa : Solve the given homogeneous equation by using an appropriate substitution.

End Chapter 2