$$y{y}^{\prime }-a(1-\frac{b}{x})y=a^{2}b$$

$$y{y}^{\prime }=x^{n-1}((2n+1)x+an)y-n{x}^{2n}(x+a)$$

$$y{y}^{\prime }=a(-bn+x)x^{n-1}y+c(x^2-(2n+1)bx+n(n+1)b^2)x^{2n-1}$$

$$y{y}^{\prime }=(a(2n+k)x^k+b)x^{n-1}y+(-a^{2}n{x}^{2k}-ab{x}^k+c)x^{2n-1}$$

$$y{y}^{\prime }=(a(2n+k)x^{2k}+b(2m-k))x^{m-k-1}y-\frac{a^{2}m{x}^{4k}+c{x}^{2k}+b^{2}m}{x}$$

$$y{y}^{\prime }=\frac{((m+2L-3)x+n-2L+3)y}{x}+((m-L-1)x^2+(n-m-2L+3)x-n+L-2)x^{1-2L}$$

$$y{y}^{\prime }=(a(2n+1)x^2+cx+b(2n-1))x^{n-2}y-(n{a}^2{x}^4+ac{x}^3+n{b}^2+bcx+d{x}^2)x^{2n-3}$$

$$y{y}^{\prime }=(a(n-1)x+b(2\lambda +n))x^{\lambda -1}{(ax+b)}^{-\lambda -2}y-(anx+b(\lambda +n))x^{-1+2\lambda }{(ax+b)}^{-2\lambda -3}$$

$$y{y}^{\prime }-\frac{a((m-1)x+1)y}{x}=\frac{a^2(mx+1)(x-1)}{x}$$

$$y{y}^{\prime }-a(1-\frac{b}{\sqrt{x}})y=\frac{a^{2}b}{\sqrt{x}}$$

$$y{y}^{\prime }=\frac{3y}{{(ax+b)}^{\frac{1}{3}}{x}^{\frac{5}{3}}}+\frac{3}{{(ax+b)}^{\frac{2}{3}}{x}^{\frac{7}{3}}}$$

$$3y{y}^{\prime }=\frac{(-7\lambda s(3s+4\lambda )x+6s-2\lambda )y}{x^{\frac{1}{3}}}+\frac{6\lambda sx-6}{x^{\frac{2}{3}}}+2(\lambda s(3s+4\lambda )x+5\lambda )(-\lambda s(3s+4\lambda )x+3s+4\lambda )x^{\frac{1}{3}}$$

$$y{y}^{\prime }+\frac{a(6x-1)y}{2x}=-\frac{a^2(x-1)(4x-1)}{2x}$$

$$y{y}^{\prime }-\frac{a(1+\frac{2b}{x^2})y}{2}=\frac{a^2(3x+\frac{4b}{x})}{16}$$

$$y{y}^{\prime }+\frac{a(13x-20)y}{14x^{\frac{9}{7}}}=-\frac{3a^2(x-1)(x-8)}{14x^{\frac{11}{17}}}$$

$$y{y}^{\prime }+\frac{5a(23x-16)y}{56x^{\frac{9}{7}}}=-\frac{3a^2(x-1)(25x-32)}{56x^{\frac{11}{17}}}$$

$$y{y}^{\prime }+\frac{a(19x+85)y}{26x^{\frac{18}{13}}}=-\frac{3a^2(x-1)(x+25)}{26x^{\frac{23}{13}}}$$

$$y{y}^{\prime }+\frac{a(13x-18)y}{15x^{\frac{7}{5}}}=-\frac{4a^2(x-1)(x-6)}{15x^{\frac{9}{5}}}$$

$$y{y}^{\prime }+\frac{a(5x+1)y}{2\sqrt{x}}=a^2(-x^2+1)$$

$$y{y}^{\prime }+\frac{3a(19x-14)x^{\frac{7}{5}}y}{35}=-\frac{4a^2(x-1)(9x-14)x^{\frac{9}{5}}}{35}$$

$$y{y}^{\prime }+\frac{3a(3x+7)y}{10x^{\frac{13}{10}}}=-\frac{a^2(x-1)(x+9)}{5x^{\frac{8}{5}}}$$

$$y{y}^{\prime }+\frac{a(7x-12)y}{10x^{\frac{7}{5}}}=-\frac{a^2(x-1)(x-16)}{10x^{\frac{9}{5}}}$$

$$y{y}^{\prime }+\frac{3a(13x-8)y}{20x^{\frac{7}{5}}}=-\frac{a^2(x-1)(27x-32)}{20x^{\frac{9}{5}}}$$

$$y{y}^{\prime }+\frac{3a(3x+11)y}{14x^{\frac{10}{7}}}=-\frac{a^2(x-1)(x-27)}{14x^{\frac{13}{7}}}$$

$$y{y}^{\prime }-\frac{a(x+1)y}{2x^{\frac{7}{4}}}=\frac{a^2(x-1)(3x+5)}{4x^{\frac{5}{2}}}$$

$$y{y}^{\prime }-\frac{a(x+1)y}{2x^{\frac{7}{4}}}=\frac{a^2(x-1)(x+5)}{4x^{\frac{5}{2}}}$$

$$y{y}^{\prime }-\frac{a(4x+3)y}{14x^{\frac{8}{7}}}=-\frac{a^2(x-1)(16x+5)}{14x^{\frac{9}{7}}}$$

$$y{y}^{\prime }+\frac{a(13x-3)y}{6x^{\frac{2}{3}}}=-\frac{a^2(x-1)(5x-1)}{6x^{\frac{1}{3}}}$$

$$y{y}^{\prime }-\frac{a(8x-1)y}{28x^{\frac{8}{7}}}=\frac{a^2(x-1)(32x+3)}{28x^{\frac{9}{7}}}$$

$$y{y}^{\prime }-\frac{a(5x-4)y}{x^4}=\frac{a^2(x-1)(3x-1)}{x^7}$$

$$y{y}^{\prime }-\frac{2a(3x-10)y}{5x^4}=\frac{a^2(x-1)(8x-5)}{5x^7}$$

$$y{y}^{\prime }+\frac{a(39x-4)y}{42x^{\frac{9}{7}}}=-\frac{a^2(x-1)(9x-1)}{42x^{\frac{11}{7}}}$$

$$y{y}^{\prime }+\frac{a(-2+x)y}{x}=\frac{2a^2(x-1)}{x}$$

$$y{y}^{\prime }+\frac{a(3x-2)y}{x}=-\frac{2a^2{(x-1)}^2}{x}$$

$$y{y}^{\prime }+\frac{a(1-\frac{b}{x^2})y}{x}=\frac{a^{2}b}{x}$$

$$y{y}^{\prime }-\frac{a(3x-4)y}{4x^{\frac{5}{2}}}=\frac{a^2(x-1)(2+x)}{4x^4}$$

$$y{y}^{\prime }+\frac{a(33x+2)y}{30x^{\frac{6}{5}}}=-\frac{a^2(x-1)(9x-4)}{30x^{\frac{7}{5}}}$$

$$y{y}^{\prime }-\frac{a(x-8)y}{8x^{\frac{5}{2}}}=-\frac{a^2(x-1)(3x-4)}{8x^4}$$

$$y{y}^{\prime }+\frac{a(17x+18)y}{30x^{\frac{22}{15}}}=-\frac{a^2(x-1)(x+4)}{30x^{\frac{29}{15}}}$$

$$y{y}^{\prime }-\frac{a(6x-13)y}{13x^{\frac{5}{2}}}=-\frac{a^2(x-1)(x-13)}{26x^4}$$

$$y{y}^{\prime }+\frac{a(24x+11)x^{\frac{27}{20}}y}{30}=-\frac{a^2(x-1)(9x+1)}{60x^{\frac{17}{10}}}$$

$$y{y}^{\prime }-\frac{2a(2+3x)y}{5x^{\frac{8}{5}}}=\frac{a^2(x-1)(8x+1)}{5x^{\frac{11}{5}}}$$

$$y{y}^{\prime }-\frac{6a(1+4x)y}{5x^{\frac{7}{5}}}=\frac{a^2(x-1)(27x+8)}{5x^{\frac{9}{5}}}$$

$$y{y}^{\prime }-\frac{a(x+4)y}{5x^{\frac{8}{5}}}=\frac{a^2(x-1)(3x+7)}{5x^{\frac{3}{5}}}$$

$$y{y}^{\prime }-\frac{a(x+4)y}{5x^{\frac{8}{5}}}=\frac{a^2(x-1)(3x+7)}{5x^{\frac{11}{5}}}$$

$$y{y}^{\prime }-\frac{a(2x-1)y}{x^{\frac{5}{2}}}=\frac{a^2(x-1)(3x+1)}{2x^4}$$

$$y{y}^{\prime }+\frac{a(x-6)y}{5x^{\frac{7}{5}}}=\frac{2a^2(x-1)(x+4)}{5x^{\frac{9}{5}}}$$

$$y{y}^{\prime }+\frac{a(21x+19)y}{5x^{\frac{7}{5}}}=-\frac{2a^2(x-1)(9x-4)}{5x^{\frac{9}{5}}}$$

$$y{y}^{\prime }-\frac{3ay}{x^{\frac{7}{4}}}=\frac{a^2(x-1)(x-9)}{4x^{\frac{5}{2}}}$$

$$y{y}^{\prime }-\frac{a((k+1)x-1)y}{x^2}=\frac{a^2(k+1)(x-1)}{x^2}$$

$$y{y}^{\prime }-a((k-2)x+2k-3)x^{-k}y=a^2(k-2){(x-1)}^2{x}^{1-2k}$$

$$y{y}^{\prime }-\frac{a((4k-7)x-4k+5)x^{-k}y}{2}=\frac{a^2(2k-3){(x-1)}^2{x}^{1-2k}}{2}$$

$$y{y}^{\prime }-((2n-1)x-an)x^{-n-1}y=n(x-a)x^{-2n}$$

$$y{y}^{\prime }-((n+1)x-an)x^{n-1}{(x-a)}^{-n-2}y=n{x}^{2n}{(x-a)}^{-2n-3}$$

$$y{y}^{\prime }-a((2k-3)x+1)x^{-k}y=a^2(k-2)((k-1)x+1)x^{2-2k}$$

$$y{y}^{\prime }-a((n+2k-3)x+3-2k)x^{-k}y=a^2((n+k-1)x^2-(n+2k-3)x+k-2)x^{1-2k}$$

$$y{y}^{\prime }-\frac{a((n+2)x-2)x^{-\frac{2n+1}{n}}y}{n}=\frac{a^2((n+1)x^2-2x-n+1)x^{-\frac{3n+2}{n}}}{n}$$

$$y{y}^{\prime }-\frac{a(\frac{(n+4)x}{n+2}-2)x^{-\frac{2n+1}{n}}y}{n}=\frac{a^2(2x^2+(n^2+n-4)x-(n-1)(n+2))x^{-\frac{3n+2}{n}}}{n(n+2)}$$

$$y{y}^{\prime }+\frac{a(\frac{(3n+5)x}{2}+\frac{n-1}{n+1})x^{-\frac{n+4}{n+3}}y}{n+3}=-\frac{a^2((n+1)x^2-\frac{(n^2+2n+5)x}{n+1}+\frac{4}{n+1})x^{-\frac{n+5}{n+3}}}{2n+6}$$

$$y{y}^{\prime }-a(\frac{n+2}{n}+b{x}^n)y=-\frac{a^{2}x(\frac{n+1}{n}+b{x}^n)}{n}$$

$$y{y}^{\prime }=(a{\mathrm{e}}^x+b)y+c{\mathrm{e}}^{2x}-ab{\mathrm{e}}^x-b^2$$

$$y{y}^{\prime }=(a(2\mu +\lambda ){\mathrm{e}}^{\lambda x}+b){\mathrm{e}}^{\mu x}y+(-a^2\mu {\mathrm{e}}^{2\lambda x}-ab{\mathrm{e}}^{\lambda x}+c){\mathrm{e}}^{2\mu x}$$

$$y{y}^{\prime }=({\mathrm{e}}^{\lambda x}a+b)y+c(a^2{\mathrm{e}}^{2\lambda x}+ab(\lambda x+1){\mathrm{e}}^{\lambda x}+b^2\lambda x)$$

$$y{y}^{\prime }={\mathrm{e}}^{\lambda x}(2a\lambda x+a+b)y-{\mathrm{e}}^{2\lambda x}(a^2\lambda x^2+abx+c)$$

$$y{y}^{\prime }={\mathrm{e}}^{ax}(2a{x}^2+b+2x)y+{\mathrm{e}}^{2ax}(-a{x}^4-b{x}^2+c)$$

$$y{y}^{\prime }+a(2bx+1){\mathrm{e}}^{bx}y=-a^{2}b{x}^2{\mathrm{e}}^{2bx}$$

$$y{y}^{\prime }-a(1+2n+2n(n+1)x){\mathrm{e}}^{(n+1)x}y=-a^{2}n(n+1)(nx+1)x{\mathrm{e}}^{2(n+1)x}$$

$$y{y}^{\prime }+a(1+2b\sqrt{x}){\mathrm{e}}^{2b\sqrt{x}}y=-a^{2}b{x}^{\frac{3}{2}}{\mathrm{e}}^{4b\sqrt{x}}$$

$$y{y}^{\prime }=(a\cosh (x)+b)y-ab\sinh (x)+c$$

$$y{y}^{\prime }=(a\sinh (x)+b)y-ab\cosh (x)+c$$

$$y{y}^{\prime }=(2\ln (x)+a+1)y+x(-\ln (x{)}^2-a\ln (x)+b)$$

$$y{y}^{\prime }=(2\ln (x{)}^2+2\ln (x)+a)y+x(-\ln (x{)}^4-a\ln (x{)}^2+b)$$

$$y{y}^{\prime }=ax\cos \left(\lambda x^2\right)y+x$$

$$y{y}^{\prime }=ax\sin \left(\lambda x^2\right)y+x$$

$$(Ay+Bx+a)y^{\prime }+By+kx+b=0$$

$$(y+ax+b)y^{\prime }=\alpha y+\beta x+\gamma$$

$$(y+ak{x}^2+bx+c)y^{\prime }=-a{y}^2+2akxy+my+k(k+b-m)x+s$$

$$(y+A{x}^n+a)y^{\prime }+nA{x}^{n-1}y+k{x}^m+b=0$$

$$(y+x^{n+1}a+b{x}^n)y^{\prime }=(an{x}^n+c{x}^{n-1})y$$

$$xy{y}^{\prime }=a{y}^2+by+c{x}^n+s$$

$$xy{y}^{\prime }=-n{y}^2+a(2n+1)xy+by-a^{2}n{x}^2-abx+c$$

$$y^{\prime \prime }+ay=0$$

$$y^{\prime \prime }-(ax+b)y=0$$

$$y^{\prime \prime }-(a^2{x}^2+a)y=0$$

$$y^{\prime \prime }-(a{x}^2+b)y=0$$

$$y^{\prime \prime }+a^{3}x(-ax+2)y=0$$

$$y^{\prime \prime }-(a{x}^2+bcx)y=0$$

$$y^{\prime \prime }-a{x}^{n}y=0$$

$$y^{\prime \prime }-a(a{x}^{2n}+n{x}^{n-1})y=0$$

$$y^{\prime \prime }-a{x}^{n-2}(a{x}^n+n+1)y=0$$

$$y^{\prime \prime }+(a{x}^{2n}+b{x}^{n-1})y=0$$

$$y^{\prime \prime }+a{y}^{\prime }+by=0$$

$$y^{\prime \prime }+a{y}^{\prime }+(bx+c)y=0$$

$$y^{\prime \prime }+a{y}^{\prime }-(b{x}^2+c)y=0$$

$$y^{\prime \prime }+a{y}^{\prime }+b(-b{x}^2+ax+1)y=0$$

$$y^{\prime \prime }+a{y}^{\prime }+bx(-b{x}^3+ax+2)y=0$$

$$y^{\prime \prime }+a{y}^{\prime }+b(-b{x}^{2n}+a{x}^n+n{x}^{n-1})y=0$$

$$y^{\prime \prime }+a{y}^{\prime }+b(-b{x}^{2n}-a{x}^n+n{x}^{n-1})y=0$$

$$y^{\prime \prime }+x{y}^{\prime }+(n-1)y=0$$

$$y^{\prime \prime }-2x{y}^{\prime }+2ny=0$$