I'm either missing something, or my book has incorrect answers...

$\displaystyle \frac{dy}{dx} = 4e^x; y(2) = \pi$

integrating gets...

$\displaystyle y = 4e^x + C$

$\displaystyle y(2) = \pi = 4e^{(2)} + C$

$\displaystyle C = \pi - 4e^{(2)} = -26.4$

Thus $\displaystyle y = 4e^x -26.4$

Is this correct? My book says $\displaystyle y = 4e^x -4 + \pi$.

Looks like they used $\displaystyle e^0$.

The other one...

$\displaystyle \frac{d^2w}{dt^2} = \sqrt{t} = t^{1/2}; w'(1) = -2, w(1) = 2$

$\displaystyle w' = \frac{2}{3}t^{3/2} + C_1$

$\displaystyle w'(1) = -2 = \frac{2}{3}(1)^{3/2} + C_1$

$\displaystyle C_1 = -\frac{8}{3}$

$\displaystyle w = \frac{4}{15}t^{5/2} - \frac{8}{3}t + C_2$

$\displaystyle w(1) = 2 = \frac{4}{15}(1)^{5/2} - \frac{8}{3}(1) + C_2$

$\displaystyle C_2 = \frac{22}{5}$

$\displaystyle w = \frac{4}{15}t^{5/2} - \frac{8}{3}t + \frac{22}{5}$

Correct?

Book says: $\displaystyle w = 4t^{5/2} - 2t +2$

-Scott