Exam 16: Exact First-Order Equations

Taylor's Theorem to find the first four terms of the series solution of $y ^ {t } - 2 x y = 0$ given the initial condition $y ( 0 ) = 1$ and use it to calculate $y ( 3 )$ . Round your answer to three decimal places wherever applicable.

$\text { Find the general solution of the differential equation } \frac { 1 } { x ^ { 2 } + y ^ { 2 } } ( x d y - 4 y d x ) = 0 \text {. }$

Find the particular solution of the differential equation $\frac { 1 } { \sqrt { x ^ { 3 } + y ^ { 3 } } } \left( x ^ { 2 } d x + y ^ { 2 } d y \right) = 0$ that satisfies the initial condition $( 8,8 )$

Find the integrating factor of the differential equation $\left( - 14 y ^ { 3 } + 2 \right) d x + \left( 21 x y ^ { 2 } + 7 x ^ { 3 } \right) d y = 0$ that is a function of $x$ or $y$ alone.

Find the general solution of the linear differential equation $y ^ { ttt} - 13 y ^ { tt } + 54 y ^ {t } - 72 y = 0 .$