Question

Solve y′′=sin(x) if y(0)=0 and y′(0)=5.

y(x)=?

Answer 1

`y(x)=6x-sin(x)`

Explanation:

Rewrite the differential equation as:

`(d^(2) y )/(dx^(2) ) =sin(x)`

Integrate both sides with respect to x:

`\int\ (d^(2) y )/(dx^(2) ) dx = \int\ sin(x) dx`

`(dy)/(dx) =-cos(x)+C_1`

Integrate one more time both sides with respect to x:

`\int\ (dy)/(dx) = \int\ -cos(x)+C_1 dx`

`y(x)=-sin(x)+C_1x+C_2`

Now that we find the solution, let's find its derivate:

`y'(x)=C_1-cos(x)`

Evaluating the initial conditions:

`y(0)=C_1(0)+C_2-sin(0)=0\\C_2=0`

`y'(0)=C_1-cos(0)=5\\C_1=5+1=6`

Replacing the value of the constants that we found in the differential equation solution:

`y(x)=6x-sin(x)`