By using a suitable substitution, bring the equation…

By using a suitable substitution, bring the equation $y-xy'=a(1+x^2y')$ to a form with separated variables, and then integrate it, given $y(1)=1$.

Step 1: Solve for $y'$

Expanding and collecting terms in $y'$: $y-a=y'x(1+ax)$, so

\[\frac{dy}{y-a}=\frac{dx}{x(1+ax)}.\]

Step 2: Partial fractions and integrate

Since $\dfrac{1}{x(1+ax)}=\dfrac1x-\dfrac{a}{1+ax}$, integrating both sides gives $\ln|y-a|=\ln\left|\dfrac{x}{1+ax}\right|+C$, so

\[y=a+\frac{Cx}{1+ax}.\]

Step 3: Apply the initial condition

With $y(1)=1$: $1=a+\dfrac{C}{1+a}$, so $C=(1-a)(1+a)=1-a^2$. Substituting back and simplifying,

\[\boxed{\,y=\dfrac{a+x}{1+ax}.}\]