Differential Equations

2020 SAJC P2 Q3 The variables $x$ and $y$ are related by the differential equation $frac{text{d}y}{text{d}x}=frac{x-y}{x-y+1}$. (i) Solve the differential equation, using the substitution $u=x-y+1$, and show that the general

2020 TMJC P1 Q8 (i) Show that the differential equation $frac{text{d}y}{text{d}x}=5left( x-y right)$ may be reduced by the substitution $w=x-y$ to $frac{text{d}w}{text{d}x}=1-5w$. Hence find the general solution for $~y$ in

2020 RVHS P2 Q2 A particle moving in a liquid is such that after $t$ seconds, its velocity is $v$ ms$^{-1}$, $vne 0$. $v$ satisfies the differential equation $vfrac{text{d}v}{text{d}t}+2{{v}^{2}}=t{{text{e}}^{-2t}}$. (i)

2020 RI P1 Q3 (i) Prove that for $x>0$, the substitution $y=ux$ reduces the differential equation $left( y-x right)left( frac{text{d}y}{text{d}x}-frac{y}{x} right)={{y}^{2}}+2{{x}^{2}}$ to $left( frac{u}{{{u}^{2}}+2}-frac{1}{{{u}^{2}}+2} right)left( frac{text{d}u}{text{d}x} right)=1$. [2] (ii) Hence

NJC Differential Equations Tutorial Q1 A particular species reproduce at the rate of $100k%$, where $k$ is a positive constant. Due to natural causes, members of the population die at

2015 SRJC P2 Q2 A tank contains water which is heated by an electric water heater working under the action of a thermostat. When the water heater is first switched

2022 TMJC P1 Q10 A squirrel falls vertically from a tall tree. The distance, $x$ metres, that the squirrel has fallen from the tree after $t$ seconds is observed. It

2022 R1 P1 Q10 Glucose in the blood stream is reduced at a rate proportional to the amount of glucose present in the blood stream.Let $G$ milligrams (mg) be the

2009 ACJC P1 Q9 The variables $x$ and $y$ are connected by the differential equation $frac{text{d}y}{text{d}x}-1-{{left( x+y right)}^{2}}=0$. (i) Show that the substitution $u=x+y$ reduces the differential equation to $frac{text{d}u}{text{d}x}-2-{{u}^{2}}=0$.