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

1. 1
   Consider the differential equation $\frac{dy}{dx}=\frac{x+y}{x}$ for $x>0$, and the substitution $y=vx$, where $v$ is a function of $x$.

   (a)

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

   [1 mark]

   • A$\frac{dv}{dx}$
   • B$x\frac{dv}{dx}$
   • C$v+\frac{dv}{dx}$
   • D$v+x\frac{dv}{dx}$

   (b)

   Which equation does the substitution produce?

   [1 mark]

   • A$x\frac{dv}{dx}=1$
   • B$\frac{dv}{dx}=1$
   • C$x\frac{dv}{dx}=1+2v$
   • D$x\frac{dv}{dx}=1+v$

   (c)

   Find the general solution, giving $y$ in terms of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the differential equation $\frac{dy}{dx}+\frac yx=y^2$ for $x>0$, $y>0$, and the substitution $z=\frac1y$.

   (a)

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

   [1 mark]

   • A$\frac{1}{y^2}\frac{dy}{dx}$
   • B$-\frac{1}{y^2}\frac{dy}{dx}$
   • C$-\frac1y\frac{dy}{dx}$
   • D$-y^2\frac{dy}{dx}$

   (b)

   Which equation does the substitution produce?

   [1 mark]

   • A$\frac{dz}{dx}+\frac zx=1$
   • B$\frac{dz}{dx}-\frac zx=1$
   • C$\frac{dz}{dx}-\frac zx=-1$
   • D$\frac{dz}{dx}+\frac zx=-1$

   (c)

   The substitution transforms the equation into $\frac{dz}{dx}-\frac zx=-1$. Find the general solution for $z$ in terms of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the differential equation $\frac{dy}{dx}=(x+y)^2$.

   (a)

   Show that the substitution $z=x+y$ transforms the equation into $\frac{dz}{dx}=1+z^2$.

   [3 marks]

   (b)

   Given that $y=0$ when $x=0$, solve the differential equation to find $y$ in terms of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Consider the differential equation $xy\frac{dy}{dx}=x^2+y^2$ for $x>0$, $y>0$.

   (a)

   Use the substitution $y=vx$ to show that $x\frac{dv}{dx}=\frac1v$, and hence find the general solution, giving $y^2$ in terms of $x$.

   [6 marks]

   (b)

   Given that $y=2$ when $x=1$, find $y$ in terms of $x$. State the values of $x$ for which this solution is valid, and find the exact value of $y$ when $x=e$.

   [6 marks]

   Total for question 4: 12 marks

End of questions