Second order equations reducible by substitutionEdexcel International A Level Further Maths: Mind map

Exam questions on Second order equations reducible by substitution

1. A differential equation for $y$ in terms of $x$ ($x>0$) is to be transformed using the substitution $x=\mathrm{e}^t$. A dot denotes differentiation with respect to $t$, so $\dot y=\frac{dy}{dt}$ and $\ddot y=\frac{d^2y}{dt^2}$.
   The equation $x^2\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+2y=0$ transforms into $\ddot y-3\dot y+2y=0$. Find the general solution for $y$ in terms of $x$.2 marks
2. The differential equation $\frac{d^2y}{dx^2}+4x\frac{dy}{dx}+(4x^2+2)y=8\mathrm{e}^{-x^2}$ is to be transformed using the substitution $y=z\,\mathrm{e}^{-x^2}$, where $z$ is a function of $x$.
   The equation transforms into $\frac{d^2z}{dx^2}=8$. Find the general solution for $y$ in terms of $x$.2 marks
3. The differential equation $x^2\frac{d^2y}{dx^2}+5x\frac{dy}{dx}+4y=0$, $x>0$, is to be solved using the substitution $x=\mathrm{e}^t$.
   Show that the substitution transforms the equation into $\frac{d^2y}{dt^2}+4\frac{dy}{dt}+4y=0$.3 marks