Answer:
[tex]y(x)=6x-sin(x)[/tex]
Step-by-step explanation:
Rewrite the differential equation as:
[tex]\frac{d^{2} y }{dx^{2} } =sin(x)[/tex]
Integrate both sides with respect to x:
[tex]\int\ \frac{d^{2} y }{dx^{2} } dx = \int\ sin(x) dx[/tex]
[tex]\frac{dy}{dx} =-cos(x)+C_1[/tex]
Integrate one more time both sides with respect to x:
[tex]\int\ \frac{dy}{dx} = \int\ -cos(x)+C_1 dx[/tex]
[tex]y(x)=-sin(x)+C_1x+C_2[/tex]
Now that we find the solution, let's find its derivate:
[tex]y'(x)=C_1-cos(x)[/tex]
Evaluating the initial conditions:
[tex]y(0)=C_1(0)+C_2-sin(0)=0\\C_2=0[/tex]
[tex]y'(0)=C_1-cos(0)=5\\C_1=5+1=6[/tex]
Replacing the value of the constants that we found in the differential equation solution:
[tex]y(x)=6x-sin(x)[/tex]
