Let U1 and U2 be two linearly independent solutions of the second-order linear differential equation d²u fu + dx² 2 - y' Let w= uz/u1. Let y=-2u₁/u₁. ,2 0, f = f(x). 2 = f. (iii) Show that w"/w' = - 2u₁/u₁, and deduce that w satisfies the equation 2 1 ()'. () - () - 2 = f.

Q: √²³₁(−12xy² + 8x³y + 3)dxdy

A: LetWe have to evaluate the above integral.

Q: y = (C₁ + C₂x) eª + sinx+x²; y" - 2y' + y = −2 cos x + x² − 4x + 2

A: Here we find the first and second derivatives of given solution and then put in the left hand side…

Q: Provide a general solution formula for the differential equation y" – 15y" + 70y" – 120y' + 64y = 0,…

A:

Q: Identify the linear differential equations. Justify and show work.  1) y''=ex 2) y''= ey 3)…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Match the differential equation with its direction field. y' = 10 - y \ 20 0:34 // 0.2f Q,1 -Q.31…

A:

Q: 2 + v - , 2+z+ 11. 0. di

A:

Q: Sketch the direction field for the differential equation. Then use it to sketch a solution curve…

A: The given differential equation is: y'=5y+25xy. We have to sketch the direction field and then use…