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Show transcribed image text Find a general solution of each reducible second-order differ- ential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11) 43. xy,-y 44. yy" + (y)2 0 55. Show that the substitution u ax + by + c transforms the differential equation dy/dx – F(ax +by + c) into a separable equation. 56. Suppose that n 0 and n夭. Show that the sub- stitution u yi-n transforms the Bernoulli equation dy/dx + P(x)y- Q(x)y" into the linear equation +(1-n)P(x)u(x) (1-n)Q(x). 57. Show that the substitution u = In y transforms the differ- ential equation dy/dx + P (x)y Q(x)(y Iny) into the linear equation dv/dx + P(x) -Q(x)v(x)
Find a general solution of each reducible second-order differ- ential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11) 43. xy,-y 44. yy" + (y)2 0 55. Show that the substitution u ax + by + c transforms the differential equation dy/dx – F(ax +by + c) into a separable equation. 56. Suppose that n 0 and n夭. Show that the sub- stitution u yi-n transforms the Bernoulli equation dy/dx + P(x)y- Q(x)y" into the linear equation +(1-n)P(x)u(x) (1-n)Q(x). 57. Show that the substitution u = In y transforms the differ- ential equation dy/dx + P (x)y Q(x)(y Iny) into the linear equation dv/dx + P(x) -Q(x)v(x)
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