Answered

IDNStudy.com, kung saan ang kuryusidad ay nagtatagpo ng kalinawan. Sumali sa aming platform ng tanong at sagot upang makakuha ng eksaktong tugon sa lahat ng iyong mahahalagang tanong.

Solve the initial value problem '' = 4 ; 2
( )=− 1, ' 2 ( )=− 1


Sagot :

Answer:

1. Integrate the differential equation:

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 ):

First, use the initial condition

[tex]( y'(2) = -1 ):[/tex]

[tex]{y'(2) = 4(2) + C_1 = -1 \implies 8 + C_1 = -1 \implies C_1 = -9}

[/tex]

Next, use the initial condition

[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]

4. Write the solution:

[tex]y = 2x^2 - 9x + 9[/tex]

Thus, the solution to the initial value problem is:

[tex]y = 2x^2 - 9x + 9[/tex]