For the linear differential equation y' + 6xy = x³e' integrating factor is: -6x2 2 the r(x) = help (formulas) After multiplying both sides by the integrating factor and "unapplying" the product rule we get the new differential equation: d dx [0] = help (formulas) Integrating both sides we get the algebraic equation = + C help (formulas) Solving for y, the general solution to the differential equation is y = ☐ help (formulas) Note: Use C as the constant. Book: Section 1.4 of Notes on Diffy Qs

Consider the differential equation y'= -2x (A) At the point (1, -1), the direction field has a slope of (B) At the point (-1, 1), the direction field has a slope of (C) Along the x-axis the slope of the direction field is equal to which number or expression? (D) Along the y-axis the slope of the direction field is equal to which number or expression? (E) Use your answers above to help choose the correct direction field for the differential equation. 1 ✓ 2 1 CAI -1- Ō 151 X / 1 / 11 1 1 / // -1 1|1 1 N Q 2 2 NN NIX V NN / / / 17 1 +5/ -17 -1 21 1 4342 V N / // 1 Q f 1 775 \ \\ \ \-1 + \\\' 1 1 1 1 1 1 1 1 2 J 1 / 1 / 1 1 1 1 7 +71 I -1 / / 1-1 / / - \ \ 1 / /-1- \ // +1 OF I 1- / 4 ¹1 111 7 1 a

y+2xy' = 2√√√x sin x solve differential

Solve y' = 6xy - 2x, 6xy y(x) = esc 7 - 2x, y(0) = 12 с ↑ C DO Ill