Second order equations reducible by substitutionEdexcel International A Level Further Maths: Subtopic test

1. 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 $\ddot y=\frac{d^2y}{dt^2}$.

   (a)

   Which expression is equal to $\frac{dy}{dx}$?

   [1 mark]

   • A$\mathrm{e}^{t}\,\dot y$
   • B$\dot y$
   • C$\mathrm{e}^{-t}\,\dot y$
   • D$\mathrm{e}^{-t}\,\ddot y$

   (b)

   Which expression is equal to $\frac{d^2y}{dx^2}$?

   [1 mark]

   • A$\mathrm{e}^{-2t}\left(\ddot y-\dot y\right)$
   • B$\mathrm{e}^{-2t}\,\ddot y$
   • C$\mathrm{e}^{-t}\,\ddot y$
   • D$\mathrm{e}^{-2t}\left(\ddot y+\dot y\right)$

   (c)

   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]

   Total for question 1: 4 marks

2. 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 $x$.

   (a)

   Which expression is equal to $\frac{dy}{dx}$?

   [1 mark]

   • A$z'\mathrm{e}^{-x^2}$
   • B$\left(z'+2xz\right)\mathrm{e}^{-x^2}$
   • C$\left(z'-x^2z\right)\mathrm{e}^{-x^2}$
   • D$\left(z'-2xz\right)\mathrm{e}^{-x^2}$

   (b)

   Which expression is equal to $\frac{d^2y}{dx^2}$?

   [1 mark]

   • A$\left(z''-4xz'+(4x^2+2)z\right)\mathrm{e}^{-x^2}$
   • B$\left(z''-4xz'+(4x^2-2)z\right)\mathrm{e}^{-x^2}$
   • C$\left(z''-2xz'+(4x^2-2)z\right)\mathrm{e}^{-x^2}$
   • D$\left(z''+4xz'+(4x^2-2)z\right)\mathrm{e}^{-x^2}$

   (c)

   The equation transforms into $\frac{d^2z}{dx^2}=8$. Find the general solution for $y$ in terms of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 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 $x=\mathrm{e}^t$.

   (a)

   Show that the substitution transforms the equation into $\frac{d^2y}{dt^2}+4\frac{dy}{dt}+4y=0$.

   [3 marks]

   (b)

   Find the solution of the original equation for which $y=1$ and $\frac{dy}{dx}=0$ when $x=1$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The differential equation $x^2\frac{d^2y}{dx^2}-3x\frac{dy}{dx}+4y=2x^2\ln x$, $x>0$, is to be solved using the substitution $x=\mathrm{e}^t$.

   (a)

   Show that the substitution transforms the equation into $\frac{d^2y}{dt^2}-4\frac{dy}{dt}+4y=2t\mathrm{e}^{2t}$, and find the complementary function in terms of $t$.

   [6 marks]

   (b)

   Find the particular solution of the original equation for which $y=1$ and $\frac{dy}{dx}=2$ when $x=1$. Give $y$ in terms of $x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions