Question

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

Answer 1

Answer:

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

Step-by-step explanation:

Rewrite the differential equation as:

$\frac{d^{2} y }{dx^{2} } =sin(x)$

Integrate both sides with respect to x:

$\int\ \frac{d^{2} y }{dx^{2} } dx = \int\ sin(x) dx$

$\frac{dy}{dx} =-cos(x)+C_1$

Integrate one more time both sides with respect to x:

$\int\ \frac{dy}{dx} = \int\ -cos(x)+C_1 dx$

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

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:

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