Past exam of the differential equations 2 course of the University of Oxford / 2014 / 3

Write the equation as $ϵx^3-(x-1{)}^2=0$. For $ϵ\to 0^{+}$, two roots of a polynomial approach the double root one. Their splitting follows a fractional-power dominant balance, rather than an ordinary expansion in integer powers.

Put $\delta =\sqrt{ϵ}$ and $x=1+a\delta +b{\delta }^2+\cdots$. Substitution givesThus $a=\pm 1$ and $b=3/2$. The two bounded roots have the Puiseux series expansionsThe first two nonzero terms are $1\pm \sqrt{ϵ}$; the displayed correction also resolves their common order-$ϵ$ shift.

The third root escapes to infinity because the leading polynomial degree drops when $ϵ=0$. Balance $ϵx^3$ against $x^2$, giving $x=O({ϵ}^{-1})$. Set $X=ϵx$; the equation becomesFor $X=1+Aϵ+B{ϵ}^2+\cdots$, the first two coefficient equations are $A+2=0$ and $B+2A^2+2A-1=0$. HenceIts first two nonzero terms are ${ϵ}^{-1}-2$. These three distinct branches account for all roots of the cubic for sufficiently small positive $ϵ$. The Vieta formulas provide a consistency check: their sum is ${ϵ}^{-1}$, with the constant and order-$ϵ$ terms cancelling.

If $ϵ$ approaches zero through negative values, the bounded pair is complex: $x_{\pm }=1\pm i\sqrt{|ϵ|}+\frac{3}{2}ϵ+O(|ϵ{|}^{3/2})$. The large-root expansion is unchanged as a real asymptotic formula. The positive-parameter regime is used for the boundary-layer problem below.

Seek an order-one outer expansion as $ϵ\to 0^{+}$:At leading order, $0=y_0-x^2$, so $y_0=x^2$. At order $ϵ$,Since $y_0^{\prime \prime }=2$, this gives $y_1=-x^4$. ThereforeThis is an approximation away from any boundary layer; the small coefficient of the highest derivative means the reduced equation cannot generally impose both endpoint conditions.

At the left endpoint, both computed outer expansion terms vanish, so $y_{\mathrm{out}}(0)=0$ agrees with the boundary data. At the right endpoint,which differs by order one from the required zero value. Hence the boundary layer is at $x=1$. The left boundary requires no order-one layer for this outer branch.

Suppose the right-end boundary layer has width $\ell (ϵ)$, and let $X=(1-x)/\ell$. An order-one change of $y$ over this width gives $ϵy^{\prime \prime }=O(ϵ/{\ell }^2)$. Near $x=1$, the leading terms $y-x^2$ are order one. The dominant balance is thereforeThusThe inner variable satisfies $X\ge 0$ inside the domain and resolves the reaction boundary layer with square-root thickness.

Write $y(x)=Y(X)$ with $X=(1-x)/\sqrt{ϵ}$. The transformed equation isSince the rescaled forcing contains $\sqrt{ϵ}$, an inner expansion naturally uses powers of $\sqrt{ϵ}$. Its leading term satisfiesThe asymptotic matching condition with the leading outer solution isThe general solution is $Y_0=1+A{e}^X+B{e}^{-X}$. Matching eliminates $A{e}^X$, and the boundary condition gives $B=-1$. HenceThis tends to the boundary value zero at $x=1$ and to the matching value one on the interior side of the layer. The negative sign in the exponential follows from measuring the inner variable inward from the right endpoint.

For an optional uniform leading approximation, combine the leading inner and outer terms and subtract their common overlap value one. The additive composite expansion isIt satisfies the right boundary condition exactly; its left-end discrepancy is the exponentially small value $-e^{-1/\sqrt{ϵ}}$. The requested leading inner expansion is $Y_0$ above.