Let be a differentiable function satisfying dg dx =x^2 ,(1+g^2) for all real

MATHEMATICS · Multi-part · CBSE · Mathematics · Differential Equations · Differential Equations

Problem

Let

g

be a differentiable function satisfying

\frac{dg}{dx}=x^2\,(1+g^2)

for all real

x

. The value of

g

x=0

g(0)=1

. (a) Obtain an explicit formula for

g(x)

defined on an interval containing

x=0

. Show every algebraic step leading to your result. (b) Using your expression from part (a), evaluate

g(1)

exactly. (c) Determine the limit

\displaystyle \lim_{x\to 0} \frac{g(x)-1}{x}

and interpret its meaning with respect to the slope of the solution curve at the origin.