For each of the following differential equations verify that the accompanying functions a solution. (i) `x(dy)/(dx)=y => y=a x` (ii) `x+y(dy)/(dx)=0 => y=+-sqrt(a^2-x^2)` (iii) `x(dy)/(dx)y+y^2 => y=a/(x+a)` (iv) `x^3(d^2y)/(dx^2)=1 => y=a x+b+1/(2x)` (v) `y=((dy)/(dx))^2 => y=1/4(x+-a)^2`

For each of the following initial value problems verify that the accompanying functions is a solution. (i) x(dy)/(dx)=1, y(1)=0 => y=logx (ii) (dy)/(dx)=y , y(0)=1 => y=e^x (iii) (d^2y)/(dx^2)+y=0, y(0)=0, y^(prime)(0)=1 => y=sinx (iv) (d^2y)/(dx^2)-(dy)/(dx)=0, y(0)=2, y^(prime)(0)=1 => y=e^x+1 (v) (dy)/(dx)+y=2, y(0)=3 => y=e^(-x)+2

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

Solve the following differential equations. 2(x-3y+1)(dy)/(dx)=(4x-2y+1)

The solution of (dy)/(dx) = (y+sqrt(x^(2) -y^(2)))/x is

(y-x(dy)/(dx))=3(1-x^(2)(dy)/(dx))

The solution of y-x((dy)/(dx))=a(y^(2)+((dy)/(dx))) is

Show that the differential equation (x-y)(dy)/(dx)=x+2y

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

Verify that the function y=e^(-3x) is a solution of the differential equation (d^(2)y)/(dx^(2))+(dy)/(dx)-6y=0

Solve the differential equation (x+2y^(2))(dy)/(dx)=y, given that when x=2,y=1