# Introduction to Differential Equations

 What is a differential equation? An equation that involves one or more derivatives of an unknown function is called a differential equation. The order of the highest derivative included in a differential equation defines the order of this equation. Examples $y \, ' = 3 x$ , the order of the highest derivative is $1$ ($y \, '$ ) so the order of this differential equation is $1$. $y \, '' + y \, ' + y = 3 x$ , the order of the highest derivative is $2$ ($y \,''$ ) so the order of this differential equation is $2$. $-2 y \, ''' + y \, '' + y^4 = 3 x$ , the order of the highest derivative is $3$ ($y \, '''$ ) so the order of this differential equation is $3$. $y = f(x)$ is a solution of a differential equation if the equation is satisfied upon substitution of $y$ and its derivatives into the differential equation. Example Verify that $y = C e^{4x} + e^{3x}$, where $c$ is a constant, is a solution to the differential equation $y \, ' - 4 y = - e^{3 x}$ $y \, '$ is given by $y \, ' = 4 C e^{4 x} + 3 e^{3 x}$ We now substitute $y \, '$ and $y$ into the left side of the equation and simplify $y \, ' - 4 y = 4 C e^{4 x} + 3 e^{3 x} - 4 (C e^{4 x} + e^{3 x})$ $= 4 C e^{4 x} + 3 e^{3 x} - 4 C e^{4 x} - 4 e^{3 x}$ $= 4 C e^{4 x} - 4 C e^{4x} + e ^{3 x}(3 - 4)$ $= - e^{3 x}$ Which is equal to the left side of the given equation and therefore $y = C e^{4 x} + e^{3 x}$ is a solution to the differential equation $y \, ' - 4 y = -e^{3 x}$. Most of the work on differential equations consists in solving these equations. For example to solve the following differential example $y \, ' = 2 x$ Let us integrate both sides of the given equation as follows $\displaystyle \int y \, ' dx = \displaystyle \int 2 x dx$ which gives $y + C_1 = x^2 + C_2$ where $C_1$ and $C_2$ are constants of integration. The solution $y$ of the above equation is given by: $y = x^2 + C$, where $C = C_2 - C-1$. More references on Differential Equations Differential Equations - Runge Kutta Method

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Updated: 2 April 2013