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

Exam questions on Differential equations reducible by substitution

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 $x$.
   Find the general solution, giving $y$ in terms of $x$.2 marks
2. Consider the differential equation $\frac{dy}{dx}+\frac yx=y^2$ for $x>0$, $y>0$, and the substitution $z=\frac1y$.
   The substitution transforms the equation into $\frac{dz}{dx}-\frac zx=-1$. Find the general solution for $z$ in terms of $x$.2 marks
3. Consider the differential equation $\frac{dy}{dx}=(x+y)^2$.
   Show that the substitution $z=x+y$ transforms the equation into $\frac{dz}{dx}=1+z^2$.3 marks