Past exam of the mathematical methods I course of the University of Oxford / 2015 resit / 9

Set $u=x+y$, so $u^{\prime }=1+y^{\prime }=1+u^2/2$. The new equation is a separable differential equation:Absorbing the constant scaling into $C$, the real solutions areEach solution is taken on a connected interval avoiding its tangent poles. No constant real solution for $u$ was lost by separation, because $1+u^2/2$ is strictly positive for real $u$.

The equation requires $x\ne 0$. Put $v=y^2$, giving $v^{\prime }=2y{y}^{\prime }$ andAn integrating factor is $x^{-4}$, so $(v{x}^{-4}{)}^{\prime }=2x^{-3}$. Integrating gives $v{x}^{-4}=C-x^{-2}$, henceReal differentiable branches require $C{x}^4-x^2>0$, with one sign on each connected solution interval. At $y=0$, the original equation would demand $0=x$, contrary to $x\ne 0$; the zeros of the radicand are therefore not regular solution points. In particular, $y\equiv 0$ is not an additional solution.

For $b\ne -1$ and initially $y\ne 0$, use the Bernoulli power substitution $v=y^{b+1}$. The equation becomesWrite $\kappa =a(b+1)$. A first-order linear differential equation then givesThe second line is the resonant case of a Bernoulli equation with exponential forcing. Recover $y$ using real branches for which the original power $y^{-b}$ is defined; for example, unrestricted noninteger powers are unambiguous on $y>0$.

The exceptional exponent $b=-1$ makes the original equation linear: $y^{\prime }=(a+e^{-2x})y$. Its solutions areDomain restrictions matter when the solution reaches zero. If $b<0$ and the real power at zero is defined, $y\equiv 0$ is also a solution. For $-1<b<0$, waiting-time solutions of a sublinear Bernoulli equation may stay zero until an arbitrary $x_0$ and then become positive, withThe joining derivative is zero because $1/(b+1)>1$. For $b=0$, the linear formula can pass through zero normally; for $b>0$, the original negative power excludes zero. These qualifications prevent dividing out a valid zero solution or inventing one where the equation is undefined.