Question

The differential equation y′′=0y′′=0 has one of the following two parameter families as its general solution: yyyy=C1ex+C2e−x=C1cos(x)+C2sin(x)=C1tan(x)+C2sec(x)=C1+C2xy=C1ex+C2e−xy=C1cos⁡(x)+C2sin⁡(x)y=C1tan⁡(x)+C2sec⁡(x)y=C1+C2x Find the solution such that y(0)=6y(0)=6 and y′(0)=9y′(0)=9.

Answer 1

`y(x)=6+9x`

Explanation:

Given differential equation,
`y''=0`

Characteristic equation is given by
`m^2=0`

`\Rightarrow m=0,0`.

Differential equation have repeated roots and solution of differential equation is
`y(x)=C_1+C_2x`.............................(1)

Initial conditions are
`y(0)=6,y'(0)=9`

Plugging first condition in equation (1),

`6=C_1+C_2(0)`

`C_1=6`

Equation (1) becomes

`y(x)=6+C_2x`............................(2)

differentiate equation (2) with respect to 'x',

`y'(x)=C_2`

Plugging second condition,

`C_2=9`

Hence,
`y(x)=6+9x`