Paper 2, Section II, B

Differential Equations | Part IA, 2014

Use the transformation

y(t)=1cx(t)dx(t)dty(t)=\frac{1}{c x(t)} \frac{d x(t)}{d t}

where cc is a constant, to map the Ricatti equation

dydt+cy2+a(t)y+b(t)=0,t>0\frac{d y}{d t}+c y^{2}+a(t) y+b(t)=0, \quad t>0

to a linear equation.

Using the above result, as well as the change of variables τ=lnt\tau=\ln t, solve the boundary value problem

dydt+y2+ytλ2t2=0,t>0y(1)=2λ\begin{gathered} \frac{d y}{d t}+y^{2}+\frac{y}{t}-\frac{\lambda^{2}}{t^{2}}=0, \quad t>0 \\ y(1)=2 \lambda \end{gathered}

where λ\lambda is a positive constant. What is the value of t>0t>0 for which the solution is singular?

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