7CDSimpleDEs


 * SImple 1 st Order DEs**

The simplest form of a Differential Equation is in the form math . \qquad \dfrac{dy}{dx}=f(x) math

To solve, take the antiderivative of both sides. math \\ . \qquad \displaystyle{\int} \dfrac{dy}{dx} \; dx = \displaystyle{ \int} f(x) \; dx \\. \\ . \qquad y = \displaystyle{ \int} f(x) \; dx \\. \\ . \qquad y = F(x) + c math

A solution with the constant of integration (**c**) is called the ** general solution **.

If we are given some specific point (x, y) then we can find the value of **c** and obtain a ** particular solution **.

Most commonly, the specific values we are given will be at the beginning or the end of the implied domain. Such specific values are called ** boundary conditions **.


 * Initial conditions ** refer to the value when time (t) = 0. {time is never negative}

Example

math \text{Solve } \dfrac{dA}{dt} = 3t+2 math given the initial condition that A = 3

Take the antiderivative of both sides with respect to **t** (since **t** is the variable of the denominator in the derivative). math . \qquad A = \displaystyle{ \int} 3t+2 \; dt math . math . \qquad A=\dfrac{3t^2}{2}+2t+c \qquad \textit{General solution} math

Initial conditions: ... ... When t = 0, A = 3, ... so c = 3

math . \qquad A=\dfrac{3t^2}{2}+2t+3 \qquad \textit{Particular solution} math

Solving DEs with Classpad

Your calculator can find both general solutions and particular solutions of DEs.

**Simple 2 nd Order DEs**

The simplest form of 2nd Order Differential Equation is in the form: math . \qquad \dfrac{d^2y}{dx^2} = f(x) math

To solve, take the antiderivative of both sides -- twice. math . \qquad y = \displaystyle{ \int \int f(x) \; dx \; dx} math

Example

math \text{Solve } f ''(x)=\sin(x) math given the conditions f(0) = 2, f '(0) = 0

Take the antiderivative of both sides with respect to x (x is the variable) math \\ . \qquad f'(x) = \displaystyle{ \int \sin(x) \; dx} \\. \\ . \qquad f'(x) = -\cos(x)+c math

When **x = 0**, **f '(x) = 0**, so **c = 1** math . \qquad f'(x)=-\cos(x)+1 math

Now integrate again math \\ . \qquad f(x)= \displaystyle{ \int -\cos(x)+1 \; dx} \\. \\ . \qquad f(x)=-\sin(x)+x+d math

When **x=0**, **f(x) = 2**, so **d = 2** math . \qquad f(x)=-\sin(x)+x+2 \qquad \textit{Particular solution} math

This example can be solved on your calculator by entering: General Solution: ... ... **dSolve(y'' = sin(x), x, y)** Particular Solution: ... ... **dSolve(y'' = sin(x), x, y, x=0, y=2, y'=0)**

or interactively by: Select **dSolve** from the **Interactive**/**Equation** menu Enter Equation: **y'' = sin(x)** inde var: **x** Depe var: **y** Condition: **x=0, y=2, y'=0**
 * Include condition**

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