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Differential equations




Differential equations

  • An equation involving derivatives of the dependent variable with respect to independent variable (variables) is known as a differential equation
  • Examples are $y\frac {dy}{dx} -x=0$, $x\frac {dy}{dx}=1 + x$

Classification of Differential Equations

1. Ordinary Differential Equations (ODEs) A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation. 2. Partial Differential Equations (PDEs) A PDE is a differential equation that contains unknown multivariable functions and their partial derivatives.

Order and Degree of a Differential Equation

Order The order of a differential equation is the highest power of the derivative present in the equation. \[ \frac{d^3y}{dx^3} + x\frac{d^2y}{dx^2} = 0 \] This equation is of third order. $\frac {dy}{dx} =e^x$ This equation is of First order. Degree The degree of a differential equation is the power of the highest order derivative when the equation is a polynomial equation in derivatives. \[ \left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0 \] This equation is of second degree.
\[ \left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + e^y= 0 \] The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined


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