Solve the different equation : `x+y(dy)/(dx)=sec(x^(2)+y^(2))`. Also find the particular solution if x=y=0.

Explanation

The given differential equation is,
`x+y(dy)/(dx)=sec(x^(2)+y^(2))` . . . (i)
Let `x^(2)+y^(2)=t`
`2x+2y(dy)/(dx)=(dt)/(dx)`
`2(x+y(dy)/(dx))=(dt)/(dx)`
`x+y(dy)/(dx)=(1)/(2)(dt)/(dx)`
From equation (i) , `(1)/(2)(dt)/(dx)=sect`
`int(dt)/(sect)=int2dx`
`intcostdt=2x+c`
`sint=2x+c`
`sin(x^(2)+y^(2))=2x+c`
When x =0, y=0, we have
`sin0=0+c` c=0
`therefore` The particular solution is given by,
`sin(x^(2)+y^(2))=2x`