Differential Equations
JEE Concepts
Order and Degree of a Differential Equation:
 The order of a differential equation is the highest order of the derivative that appears in the equation.
 The degree of a differential equation is the highest power to which the highestorder derivative is raised.
Linear and Nonlinear Differential Equations:
 A differential equation is linear if it can be written in the form: $$a_ny^{(n)}+a_{n1}y^{(n1)}+…+a_1y’+a_0y=b$$ where (a_n\ne 0) and (b) are constants.
 A differential equation is nonlinear if it cannot be written in the form of a linear differential equation.
Homogeneous and Nonhomogeneous Differential Equations:
 A differential equation is homogeneous if it can be written in the form: $$a_ny^{(n)}+a_{n1}y^{(n1)}+…+a_1y’+a_0y=0$$ where (a_n, a_{n1}, …, a_1, a_0) are constants.
 A differential equation is nonhomogeneous if it cannot be written in the form of a homogeneous differential equation.
Exact Differential Equations and Integrating Factors:
 An exact differential equation is a differential equation that can be written in the form: $$M(x,y)dx+N(x,y)dy=0$$ where (M(x,y)) and (N(x,y)) have continuous firstorder partial derivatives in some region.
 An integrating factor for an exact differential equation is a function (I(x,y)) such that the product (I(x,y)M(x,y)dx + I(x,y)N(x,y)dy) is an exact differential.
Separation of Variables:
 The method of separation of variables can be used to solve some firstorder differential equations.
 To solve a firstorder differential equation using separation of variables, we write the equation in the form: $$\frac{dy}{dx}=f(x)g(y)$$ and then integrate both sides of the equation.
Linear Differential Equations of First Order and First Degree:
 A linear differential equation of first order and first degree is an equation that can be written in the form: $$\frac{dy}{dx}+P(x)y=Q(x)$$ where (P(x)) and (Q(x)) are continuous functions in some interval.
 This type of differential equation can be solved using an integrating factor.
Bernoulli’s Differential Equation:
 A Bernoulli differential equation is a differential equation that can be written in the form: $$\frac{dy}{dx}+P(x)y=Q(x)y^n$$ where (n) is a constant.
 This type of differential equation can be transformed into a linear differential equation by making the substitution (v=y^{1n}).
Riccati’s Differential Equation:
 A Riccati differential equation is a differential equation that can be written in the form: $$\frac{dy}{dx}=P(x)y^2+Q(x)y+R(x)$$ where (P(x), Q(x)), and (R(x)) are continuous functions in some interval.
 This type of differential equation is difficult to solve in general, but there are some special cases that can be solved using a variety of techniques.
Exact Differential Equations
 An exact differential equation is one for which the integrating factor is (1).
 To check if a differential equation is exact, we first compute its total differential. If it is equal to zero, then the equation is exact.
Linear Differential Equations of Higher Order with Constant Coefficients:
 A linear differential equation of higher order with constant coefficients is an equation that can be written in the form: $$a_ny^{(n)}+a_{n1}y^{(n1)}+…+a_1y’+a_0y=0$$ where (a_n, a_{n1}, …, a_1, a_0) are constants.
 This type of differential equation can be solved using the method of undetermined coefficients or the method of variation of parameters.
Variation of Parameters:
 Variation of parameters is a method for solving nonhomogeneous linear differential equations of higher order with constant coefficients.
 The method involves replacing the constants in the homogeneous solution with functions of (x) and then solving the resulting system of differential equations.
CauchyEuler Equation:
 A CauchyEuler equation is a linear differential equation of second order with variable coefficients that can be written in the form: $$x^2y’’+ax y’ +by=0$$ where (a) and (b) are constants.
 This type of differential equation can be solved by making the substitution (x=e^t).
Legendre’s Equation:
 Legendre’s equation is a linear differential equation of second order with variable coefficients that can be written in the form: $$(1x^2)y’’  2xy’+\nu (\nu + 1)y=0$$ where (\nu) is a constant.
 This type of differential equation is important in the study of spherical harmonics.
Bessel’s Equation:
 Bessel’s equation is a linear differential equation of second order with variable coefficients that can be written in the form: $$x^2y’’+xy’+(x^2\nu^2)y=0$$ where (\nu) is a constant.
 This type of differential equation is important in the study of Bessel functions.
Frobenius Method:
 The Frobenius method is a method for solving linear differential equations with regular singular points.
 The method involves making a series expansion of the solution around the singular point and then determining the coefficients of the series.
Power Series Solutions:

Power series solutions are solutions to differential equations that can be expressed as a power series.

Power series solutions are often used to solve differential equations that have regular singular points.
**CBSE Concepts**
Order and Degree of a Differential Equation:
 The order of a differential equation is the highest order of the derivative that appears in the equation.
 The degree of a differential equation is the highest power to which the highestorder derivative is raised.
Linear and Nonlinear Differential Equations:
 A differential equation is linear if it can be written in the form: $$a_ny^{(n)}+a_{n1}y^{(n1)}+…+a_1y’+a_0y=b$$ where (a_n\ne 0) and (b) are constants.
 A differential equation is nonlinear if it cannot be written in the form of a linear differential equation.
Homogeneous and Nonhomogeneous Differential Equations:
 A differential equation is homogeneous if it can be written in the form: $$a_ny^{(n)}+a_{n1}y^{(n1)}+…+a_1y’+a_0y=0$$ where (a_n, a_{n1}, …, a_1, a_0) are constants.
 A differential equation is nonhomogeneous if it cannot be written in the form of a homogeneous differential equation.
Exact Differential Equations and Integrating Factors:
 An exact differential equation is a differential equation that can be written in the form: $$M(x,y)dx+N(x,y)dy=0$$ where (M(x,y)) and (N(x,y)) have continuous firstorder partial derivatives in some region.
 An integrating factor for an exact differential equation is a function (I(x,y)) such that the product (I(x,y)M(x,y)dx + I(x,y)N(x,y)dy) is an exact differential.
Separation of Variables:
 The method of separation of variables can be used to solve some firstorder differential equations.
 To solve a firstorder differential equation using separation of variables, we write the equation in the form: $$\frac{dy}{dx}=f(x)g(y)$$ and then integrate both sides of the equation.
Linear Differential Equations of First Order and First Degree:
 A linear differential equation of first order and first degree is an equation that can be written in the form: $$\frac{dy}{dx}+P(x)y=Q(x)$$ where (P(x)) and (Q(x)) are continuous functions in some interval.
 This type of differential equation can be solved using an integrating factor.
Bernoulli’s Differential Equation:
 A Bernoulli differential equation is a differential equation that can be written in the form: $$\frac{dy}{dx}+P(x)y=Q(x)y^n$$ where (n) is a constant.
 This type of differential equation can be transformed into a linear differential equation by making the substitution (v=y^{1n}).
Riccati’s Differential Equation:
 A Riccati differential equation is a differential equation that can be written in the form: $$\frac{dy}{dx}=P(x)y^2