Find a particular solution that satasfies the D.E f "(x)= 2x and its initial conditions f '(1)= 2, f(0)= -1

zzzzz
Aug 4, 2017

#1**+2 **

f " (x) = 2x f ' (1) = 2, f (0) = -1

f "(x) = 2x integrate to find the first derivative

f ' (x) = x^2 + C_{1 } applying the initial condition to find C_{1} , we have that

f '( 1) = 2

2 = (1)^2 + C_{1}

2 = 1 + C_{1}

1 = C_{1}

So.... f'(x) = x^2 + 1

Integrate this again to find y(x)

f(x) = (1/3)x^2 + x + C_{2}

And applying the second initial condition to find C_{2}, we have that

f(0) = -1

-1 = (1/3)(0)^3 + 0 + C_{2}

C_{2} = -1

So..... f(x) = (1/3)x^3 + x - 1

CPhill
Aug 4, 2017