IBDP Past Year Exam Questions – Differential Equations

1.   [M09/P3/TZ0]

Consider the differential equation  $\frac{dy}{dx}=\frac{y^2 + x^2}{2x^2}$ for which $y=–1$ when $x=1$ .

(a)   Use Euler’s method with a step length of 0.25 to find an estimate for the value of  $y$ when  $x=2$ .   [7 marks]

(b)   (i)   Solve the differential equation giving your answer in the form $y=f\left(x\right)$ .

        (ii)  Find the value of  $y$ when  $x=2$ .   [13 marks]

2.   [N09/P3/TZ0]

Solve the differential equation  $\frac{dy}{dx}=\frac{y}{x} + \frac{y^2}{x^2}$ (where  $x>0$ )

given that  $y=2$ when  $x=1$ . give your answer in the form  $y=f\left(x\right)$ .   [13 marks]

3. [M10/P3/TZ0]

Given that  $\frac{dy}{dx}–2y^2=e^x$  and  $y=1$ when  $x=0$ , use Euler’s method with a step length of 0.1 to find an approximation for the value of  $y$ when  $x=0.4$ . Give all intermediate values with maximum possible accuracy.

4. [N10/P3/TZ0]

Solve the differential equation  $\left(x–1\right)\frac{dy}{dx} + xy=\left(x–1\right)e^{–x}$

given that  $y=1$ when  $x=0$ . Give your answer in the form  $y=f\left(x\right)$ .    [13 marks]