IDNStudy.com, ang iyong mapagkukunan ng mabilis at pangkomunidad na mga sagot. Makakuha ng mga sagot sa iyong mga tanong mula sa aming mga eksperto, handang magbigay ng mabilis at tiyak na solusyon.
Answer:
Since ( y'' = 4 ), we integrate with respect to ( x ) to find( y' ):
[tex]y'' = 4 \implies y' = 4x + C_1[/tex]
Here, ( C_1 ) is the constant of integration.
2. Integrate again to find ( y ):
Now, integrate ( y' ) with respect to ( x ) to find ( y ):
[tex]{y' = 4x + C_1 \implies y = 2x^2 + C_1 x + C_2}[/tex]
Here, ( C_2) is another constant of integration.
3. Use the initial conditions to find the constants ( C_1 ) and ( C_2 ):
[tex]( y'(2) = -1 ):[/tex]
[tex]{y'(2) = 4(2) + C_1 = -1 \implies 8 + C_1 = -1 \implies C_1 = -9}
[/tex]
[tex]( y(2) = -1 ):[/tex]
[tex]{y(2) = 2(2)^2 + (-9)(2) + C_2 = -1 \implies 8 - 18 + C_2 = -1 \implies -10 + C_2 = -1 \implies C_2 = 9}[/tex]
[tex]y = 2x^2 - 9x + 9[/tex]
Thus, the solution to the initial value problem is:
[tex]y = 2x^2 - 9x + 9[/tex]