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**LinearAndNon-LinearDifferentialEquations **concept and as well as few questions and their solutions to get complete idea for all readers.

**LinearAndNon-LinearDifferentialEquations** :

**Linear Differential Equation** :

An nth-order differential equation is said to be Linear Differential Equation if F is linear in y, y', y"......\(y^{(n)}\) .

General form of nth order linear differential equation is

\(a_{n}(x)y^{n}\) + \(a_{n-1}(x)y^{n-1}\) +............\(a_{1}(x)y'\) +\(a_{0}(x)y\)=g(x) ..................(1)

In equation (1) if n=1 it is called as linear first order differential equation, and if n=2 second order linear differential equation.

General form of linear first order differential equation is

\(a_{1}(x)y'\) + \(a_{0}xy\)=g(x) .................(2)

Linear second order differential equation is \(a_{2}(x)y"\) + \(a_{1}(x)y'\) + \(a_{0}(x)y\) = g(x). .................(3)

Two important properties of Linear Ordinary Differential Equation in above equation (1) are

i) The dependent variable y and all its derivatives y, y', y"......\(y^{(n)}\) are of the first degree i.e. the power of each term involving y is 1.

ii) The coefficients \(a_{0}\) , \(a_{1}\) , \(a_{2}\), ....\(a_{n}\) of y, y', y"......\(y^{(n)}\) depend at most on the independent variable x.

Non-Linear Differential equation :

A nonlinear ordinary differential equation is simply one that is not linear. Nonlinear functions of the dependent variable or its derivatives, such as sin y or \(e^{y'}\), cannot

appear in a linear equation.**Examples **

** LinearAndNon-LinearDifferentialEquations** :

1) (x+y)dy + xydx =0 is first order linear differential equation.

2) y" + 3y' + 2y = 0 is second order linear differential equation.

3) y"' + 3y" + 3y' +y =0 is third order linear differential equation .

Examples of **Non-Linear differential equations**:

1) (1+2y)y'+ 3y = sin(x) is first order non-linear differential equation .

2) y" + sin(y) =0 is second order non-linear differential equation

3) y"' + \(y^{2}\) =0 is third order non linear differential equation .

QUIZ

I) State the * order* of the given ordinary differential

equation. Determine whether the equation is

1) (1-x)y" - 5xy' + 5y = sin(x)

2) xy"' + \((y')^{4}\) + y=0

3) \(x^{5}y^{(3)}- x^{3}{y}''+6y\) = 0

4)\({y}''=\sqrt{1+({y}')^{2}}\)

5)\((sin\theta) {y}'' - (cos\theta) {y}' = 3\)

II) determine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equation.

i) \((y^{2}-1)\)dx + x dy = 0 ; in y; in x .

ii) udv + (v+uv- u\(e^{u}\))du =0 ; in v; in u.

**Solutions:**

Solutions of LinearAndNon-LinearDifferentialEquations above Quiz as follows : Please check your solutions .

I)

1) Order =2 ; Linear Differential Equation .

2) Order = 3 ; Non-Linear Differential Equation .

3) Order = 3 ; Linear Differential Equation .

4) Order = 2 ; Non-Linear Differential Equation.

5) Order = 2 ; Linear Differential Equation .

II)

i)Depended variable y is

\(frac{dy}{dx}\) = \(frac{1-y^{2}}{x}\)

x y' + \(y^{2}\)=1 , which is not linear .

Depended in x is \(frac{dx}{dy}\) = \(frac{x}{1-y^{2}}\)

\((1-y^{2}\)x' - x= 0 which is linear .

ii) In v , uv' + (1+u)v= u\(e^{u}\) which is linear in v.

In u , (v+uv-u\(e^{u}\))u' + u = 0 , which is not linear in u.

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