**Order of a
differential equation **

**Order of a
differential equation is defined as the order of the highest order derivative of
the dependent variable with respect to the independent variable involved in the
given differential equation.****\[Eqn.1)..\frac{dy}{dx}+y=0\Rightarrow order\: of\: the\: D.E.\: is\: \: one\]\[Eqn.2)..\left (\frac{d^{2}y}{dx^{2}} \right )^{5}+y=0 \Rightarrow order\: of\: the\: D.E.\: is\: \: two\]\[Eqn.3)..\left (\frac{d^{3}y}{dx^{3}} \right )^{2}+\frac{d^{4}y}{dx^{4}}+y=0\Rightarrow order\: of\: the\: D.E.\: is\: \: four\]**

**Degree of differential equation**

**It is defined as the degree(or power) of the highest order derivative involved in the given differential equation.**

**Degree of a differential equation is defined only if the differential equation is the polynomial equation in y'.**

**Order of every D.E. exist but degree may or may not be exist.**

**Order and degree if exist are always positive.**

**Degree of differential eqn.1) is one.**

**Degree of differential eqn.2) is five.**

**Degree of differential eqn.3) is one.**

**Degree of the following differential equations is not exist.\[\frac{dy}{dx}+sin\left (\frac{dy}{dx} \right )+1\]\[\frac{d^{3}y}{dx^{3}}+log\left (\frac{dy}{dx} \right )+1\] \[y'''+y^{2}+e^{y'}\]In all these differential equations we can find the order but degree is not exist. Because these equations are not the polynomial equations in its derivatives.\[\left (\frac{d^{3}y}{dx^{3}} \right )^{4}+\left ( \frac{dy}{dx} \right )^{2}+e^{\frac{dy}{dx}}+5=0\]In the above equation order is 3 but degree is not exist.**

**General Solution and ****Particular**** Solution of the differential equation.**

**Value of dependent variable(y) which satisfy the given differential equation is called solution of that D. E.**

**Solutions are of two types**

**1) General Solution:**

**Solution with the arbitrary constants a, b, c.....etc. is called the general solution of the differential equation.**

**2) Particular Solution:**

**Solution of the D.E. obtained after eliminating the arbitrary constants is called particular solution.**

**Note: Number of arbitrary constants is equal to the order of the differential equation.**

**For example:**

** If order of the differential equation is two then number of arbitrary constants is also two.**

**If order of the differential equation is five then number of arbitrary constants is also five.**

**Different Methods of Solving Differential Equations**

**There are mainly three methods of solving Differential Equations**

**1) Variable Separable Method**

**2) Homogeneous Differential Equations.**

**3) Linear Differential Equations.**

**Explanations of Methods of solving Differential Equations**

**1) Variable Separable Method**

**i) Write the differential equation.**

**ii) Bring all the terms with variable y to the LHS and bring all the terms with variable x to the RHS.**

**iii) Bring the negative sign if any to the RHS.**

**iv) Integrating on both sides and get General Solution.**

**v) For particular solution putting the given values of x and y in the General Solution and find the value of C (Constant).**

**vi) Putting the value of C in the general solution to get the particular solution. **

**2) Homogeneous Differential Equations**

**If dy/dx=F(x, y) is a differential equation, then it is called a homogeneous differential equation if it can be put in the form f(y/x) of degree zero.**

**In other words****\[If\:\; \frac{dy}{dx}=F(x,y)=f\left (\frac{y}{x} \right ) \; of\; order\; \; zero\]****Then F(x, y) is called the ****homogeneous**** differential equation.**

**Method of solving H.D.E.**

**i) Put y/x = v ⇒ y = vx**

**ii) Differentiating w.r.t. x on both side we get ****\[\frac{dy}{dx}=v+x\frac{dv}{dx}\]**

**iii) Putting the values from eqn.(i) and eqn.(ii) in the given homogeneous differential equation.**

**iv) Now separate the variable v and x.**

**v) Integrating on both side w. r. t. x we get the value of v in terms of x and arbitrary constants.**

**vi) Putting v = y/x and then find the value of y. This is called the general solution of H.D.E.**

**vii). To find the particular solution from General solution we eliminate the arbitrary constants.**

**Other forms of Homogeneous Differential Equation**

**Other form of HDE is given as**\[\frac{dx}{dy}=F(x,y)= f\left ( \frac{x}{y} \right )\]In such type of equations we put x/y = v **⇒ x = vy and then differentiating w.r.t y on both side. Then follow the sequence of steps as given above to find the value of x.**

**3) Linear Differential Equation(LDE)**

**An equation of the following type is called Linear Differential Equation.\[\frac{dy}{dx}+Py=Q\]Where P and Q are either constants or the functions of x only and the coefficient of dy/dx should be unity.**

**Method of solving LDE**

**i) Make the coefficient of dy/dx unity (or 1) and then check whether it is a LDE or not.**

**ii) Find the Integrating Factor (IF)\[IF=e^{\int Pdx}\]**

**iii) Multiplying on both side by IF****\[IF\times \frac{dy}{dx}+IF\times y=IF\times Q\]\[\frac{d}{dx}\left ( y\times IF \right )=Q\times IF\]**

**iv) Integrating on both side we get\[\int \frac{d}{dx}\left ( y\times IF \right )dx=\int \left (Q\times IF \right )dx+C\]\[y\times IF=\int \left (Q\times IF \right )dx+C\]From this equation we can find the value of y.**

**Other form of Linear differential equation\[\frac{dx}{dy}+Px=Q\] Where P and Q are either constants or the functions of y only. In this case Integrating Factor(IF) can be calculated as \[I.F.=e^{\int Pdy}\]The solution of LDE is given as \[x\times IF=\int \left (Q\times IF \right )dy+C\]**

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