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MATHMAA

Linear And Non-LinearDifferentialEquations

LinearAndNon-LinearDifferentialEquations

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

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 linear or
nonlinear .

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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