Differential equations reducible by substitutionEdexcel International A Level Further Maths: Flashcards

What these 13 flashcards ask

What is \frac{dy}{dx} if y=vx?
What type of equation is \frac{dy}{dx}=\frac{x^2+y^2}{xy}?
Result of y=vx on \frac{dy}{dx}=\frac{x+y}{x}?
What is \frac{dz}{dx} if z=x+y?
What is \frac{dz}{dx} if z=y-x?
What is \frac{dz}{dx} if z=\frac1y?
What is \frac{dz}{dx} if z=y^2?
Standard integral: \int\frac{1}{1+z^2}\,dz?
Last step when using a substitution?
After y=vx, how are the variables separated?
Why divide by y^2 when z=\frac1y?
What should you state for solutions with \ln x or \sqrt{\ \ }?
After using z=\frac1y the equation is \frac{dz}{dx}-\frac zx=-1. The integrating factor?

Exam questions on Differential equations reducible by substitution

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
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
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

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).