14quad y_(1)x=at^(2),quad y=2at-th_(m)(dy_(y))/(dx^(2))

y^2+x^2(dy)/(dx)=x y(dy)/(dx)

(dy)/(dx)=(y^(2)+y+1)/(x^(2)+x+1)

(dy)/(dx)=(y^(2)-y+1)/(x^(2)-x+1)

For each of the following differential equations verify that the accompanying functions a solution.Differential Function x(dy)/(dx)=yy=axx+y(dy)/(dx)=0y=+-sqrt(a^(2)-x^(2))x(dy)/(dx)y=y^(2)y=(a)/(x+a)x^(3)(d^(2)y)/(dx^(2))=1y=ax+b+(1)/(2x)y=((dy)/(dx))^(2)y=(1)/(4)(x+-a)^(2)

(dy)/(dx)=2((y+2)^(2))/((x+y-1)^(2))

If x^(2)+y^(2)=1 then (y'=(dy)/(dx),y''=(d^(2)y)/(dx^(2)))

If y={x +sqrt(x^(2)+1)}^(m) , then show that (x^(2)+1)(d^(2)y)/(dx^(2))+x(dy)/(dx)-m^(2)y=0.

If y=e^(m sin ^(-1)x) , then show that (1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)-m^(2)y=0 .

Verify that y=e^(m sin^(-1)x) is a solution of the differential equation (1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)-m^(2)y=0.

Verify that y=e^(m cos^(-1)x) satisfies the differential equation (1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)-m^(2)y=0