`=>` In general, an equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation.

`=>` A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation, e.g.,

`color{orange} {2 (d^y)/(dx^2) + ((dy)/(dx) )^3 = 0}` is an ordinary differential equation.

`=>` A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation, e.g.,

`color{orange} {2 (d^y)/(dx^2) + ((dy)/(dx) )^3 = 0}` is an ordinary differential equation.

Order of a differential equation :

Degree of a differential equation :

Degree of a differential equation :

`color{red}{=>"The solution which contains arbitrary constants is called the general solution (primitive) of the differential equation."}`

`color{red}{=>" The solution free from arbitrary constants i.e., the solution obtained from the general solution"}`

`color{red}{" by giving particular values to the arbitrary constants is called a particular solution of the differential equation."}`

`color{red}{=>" The solution free from arbitrary constants i.e., the solution obtained from the general solution"}`

`color{red}{" by giving particular values to the arbitrary constants is called a particular solution of the differential equation."}`

(i) By variable separable method : separate the variable with `dx` and `dy` and solve the eqaution

(ii) Solution of homogeneous differential equation :

(ii) Solution of homogeneous differential equation :

(i) Write the given differential equation in the form `(dy)/(dx) + Py = Q` where where P, Q are constants or functions of x only.

(ii) Find the Integrating Factor (I.F) `= color {red} {e^(int Pdx)}`

(iii) Write the solution of the given differential equation as

`color {red} {y (I.F) = ∫ (Q × I.F)dx + C}`

`\color{green} ✍️` In case, the first order linear differential equation is in the form `(dx)/(dy) + P_1x =Q_1`

`=>` where, `P_1` and `Q_1` are constants or functions of y only. Then `I.F =e^( ∫ P_1 dy )` and the solution of the differential equation is given by

`color {red} {x . (I.F) = ∫ (Q_1 × I.F ) dy + C}`

(ii) Find the Integrating Factor (I.F) `= color {red} {e^(int Pdx)}`

(iii) Write the solution of the given differential equation as

`color {red} {y (I.F) = ∫ (Q × I.F)dx + C}`

`\color{green} ✍️` In case, the first order linear differential equation is in the form `(dx)/(dy) + P_1x =Q_1`

`=>` where, `P_1` and `Q_1` are constants or functions of y only. Then `I.F =e^( ∫ P_1 dy )` and the solution of the differential equation is given by

`color {red} {x . (I.F) = ∫ (Q_1 × I.F ) dy + C}`