Differential equations — Question 84

ANSWER

(b) $y=e^{x^2/2}-1$; (c) $y(1)=0.649$.

Worked solution

(a) The solution passes through $(0,0)$ with a horizontal tangent. It is symmetric about the $y$-axis and rises on both sides of the origin, following the slope field.

(b) Separate variables and integrate: $$\frac{dy}{1+y}=x\,dx\quad\Rightarrow\quad\ln|1+y|=\frac{x^2}{2}+C.$$ Using $y(0)=0$ gives $C=0$. The branch through the origin has $1+y>0$, so $$y=e^{x^2/2}-1.$$

(c) $y(1)=e^{1/2}-1\approx0.649$.

Mark allocation

Draws a reasonable solution curve.

[1 mark]

Separates and integrates.

[2 marks]

Obtains the particular solution.

[1 mark]

Calculates the numerical value.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.