Tell whether the following initial value problems have a solution or not, and if its solution is unique.
1) y ′ = ( 12 x 2 + 5 x ) ( y + 9 y 3 ) , y ( 7 ) = 11 {\displaystyle y'=(12x^{2}+5x)(y+9y^{3}),y(7)=11}
2) y ′ = l n ( 7 x ) , y ( − 1 ) = 10 {\displaystyle y'=ln(7x),y(-1)=10}
3) y ′ = x + 7 x 2 − 6 x 3 y 2 − 1 , y ( 0 ) = 16 {\displaystyle y'={\frac {x+7x^{2}-6x^{3}}{y^{2}-1}},y(0)=16}
4) y ′ = x l n ( y − 1 ) , y ( 1 ) = 1 {\displaystyle y'=xln(y-1),y(1)=1}
5) y ′ = x 3 + 5 x y 2 + 7 y + 12 , y ( 5 ) = 9 {\displaystyle y'={\frac {x^{3}+5x}{y^{2}+7y+12}},y(5)=9}
6) y ′ = y + 7 y 2 x − 5 , y ( 5 ) = 4 {\displaystyle y'={\frac {y+7y^{2}}{x-5}},y(5)=4}
7) y ′ = y 3 s e c 2 ( x ) , {\displaystyle y'=y^{3}sec^{2}(x),}
8) y ′ = 5 y 2 + 6 y {\displaystyle y'={\frac {5y^{2}+6}{y}}}
9) y ′ = x 3 / y 3 {\displaystyle y'=x^{3}/y^{3}}
10) y ′ = x 2 + 3 x − 9 {\displaystyle y'=x^{2}+3x-9}
11) y ′ = c o s ( y ) / s i n ( y ) {\displaystyle y'=cos(y)/sin(y)}
12) y ′ = c o s ( x ) s i n ( y ) {\displaystyle y'={\frac {cos(x)}{sin(y)}}}
13) y ′ = c o s ( x ) + s i n ( x ) , y ( 0 ) = 1 {\displaystyle y'=cos(x)+sin(x),y(0)=1}
14) y ′ = 7 y 2 , y ( 5 ) = 9 {\displaystyle y'=7y^{2},y(5)=9}
