1. Find a particular solution of the differential equation and calculate the value of the resulting function y = φ (x) at x = x0 to the nearest two decimal places.
1.16 y´´ = x / e2x, x0 = -1/2, y (0) = 1/4, y´(0) = -1/4.
2. Find the general solution of the differential equation that admit a lowering of the order
2.16 y´´ + 2xy´2 = 0
3. Solve the Cauchy problem for differential equations admitting a reduction of order.
3.16 y´´ + 2 / (1-y) y´2 = 0, y (0) = 0, y´(0) = 1.
4. Integrate the following equation.
4.16 (xdx + ydy) / √ (x2 + y2) + (xdy-ydx) / x2 = 0
5. Record equation of the curve, passing through the point A (x0, y0), and possessing a the following property: length the perpendicular dropped from the origin of coordinates onto the tangent to the curve, is equal to the abscissa the point of tangency.
5.16 A (-4, 1)
Detailed solution. Decorated in Microsoft Word 2003. (Target decided to use formula editor)
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