1.08k Separable differential equations: dy/dx = f(x)g(y)

8 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = ( x + 2 ) \sin ^ { 2 } 2 \theta$$ and it is given that \(x = 0\) when \(\theta = 0\). Solve the differential equation and calculate the value of \(x\) when \(\theta = \frac { 1 } { 4 } \pi\), giving your answer correct to 3 significant figures.

10 Naturalists are managing a wildlife reserve to increase the number of plants of a rare species. The number of plants at time \(t\) years is denoted by \(N\), where \(N\) is treated as a continuous variable.

1. It is given that the rate of increase of \(N\) with respect to \(t\) is proportional to ( \(N - 150\) ). Write down a differential equation relating \(N , t\) and a constant of proportionality.
2. Initially, when \(t = 0\), the number of plants was 650 . It was noted that, at a time when there were 900 plants, the number of plants was increasing at a rate of 60 per year. Express \(N\) in terms of \(t\).
3. The naturalists had a target of increasing the number of plants from 650 to 2000 within 15 years. Will this target be met?

10 A large field of area \(4 \mathrm {~km} ^ { 2 }\) is becoming infected with a soil disease. At time \(t\) years the area infected is \(x \mathrm {~km} ^ { 2 }\) and the rate of growth of the infected area is given by the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k x ( 4 - x )\), where \(k\) is a positive constant. It is given that when \(t = 0 , x = 0.4\) and that when \(t = 2 , x = 2\).

1. Solve the differential equation and show that \(k = \frac { 1 } { 4 } \ln 3\).
2. Find the value of \(t\) when \(90 \%\) of the area of the field is infected.

5 \includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-2_346_437_1155_854} The diagram shows a variable point \(P\) with coordinates \(( x , y )\) and the point \(N\) which is the foot of the perpendicular from \(P\) to the \(x\)-axis. \(P\) moves on a curve such that, for all \(x \geqslant 0\), the gradient of the curve is equal in value to the area of the triangle \(O P N\), where \(O\) is the origin.

1. State a differential equation satisfied by \(x\) and \(y\). The point with coordinates \(( 0,2 )\) lies on the curve.
2. Solve the differential equation to obtain the equation of the curve, expressing \(y\) in terms of \(x\).
3. Sketch the curve.

5 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( x + 2 )$$ and it is given that \(y = 2\) when \(x = 1\). Solve the differential equation and obtain an expression for \(y\) in terms of \(x\).

5 The coordinates \(( x , y )\) of a general point on a curve satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 2 - x ^ { 2 } \right) y$$ The curve passes through the point \(( 1,1 )\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\).

4 The number of insects in a population \(t\) weeks after the start of observations is denoted by \(N\). The population is decreasing at a rate proportional to \(N \mathrm { e } ^ { - 0.02 t }\). The variables \(N\) and \(t\) are treated as continuous, and it is given that when \(t = 0 , N = 1000\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = - 10\).

1. Show that \(N\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = - 0.01 \mathrm { e } ^ { - 0.02 t } N .$$
2. Solve the differential equation and find the value of \(t\) when \(N = 800\).
3. State what happens to the value of \(N\) as \(t\) becomes large.

6 The variables \(x\) and \(\theta\) satisfy the differential equation $$\sin \frac { 1 } { 2 } \theta \frac { d x } { d \theta } = ( x + 2 ) \cos \frac { 1 } { 2 } \theta$$ for \(0 < \theta < \pi\). It is given that \(x = 1\) when \(\theta = \frac { 1 } { 3 } \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\cos \theta\).

9 The variables \(x\) and \(t\) satisfy the differential equation \(5 \frac { \mathrm {~d} x } { \mathrm {~d} t } = ( 20 - x ) ( 40 - x )\). It is given that \(x = 10\) when \(t = 0\).

1. Using partial fractions, solve the differential equation, obtaining an expression for \(x\) in terms of \(t\). [9]
2. State what happens to the value of \(x\) when \(t\) becomes large.

8 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = ( x + 2 ) \sin ^ { 2 } 2 \theta$$ and it is given that \(x = 0\) when \(\theta = 0\). Solve the differential equation and calculate the value of \(x\) when \(\theta = \frac { 1 } { 4 } \pi\), giving your answer correct to 3 significant figures.

8 A certain curve is such that its gradient at a point \(( x , y )\) is proportional to \(\frac { y } { x \sqrt { x } }\). The curve passes through the points with coordinates \(( 1,1 )\) and \(( 4 , \mathrm { e } )\).

1. By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).
2. Describe what happens to \(y\) as \(x\) tends to infinity.

7 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 1 } { ( x + 1 ) ( x + 3 ) }$$ It is given that \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

10 The variables \(x\) and \(t\) satisfy the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = x ^ { 2 } ( 1 + 2 x )\), and \(x = 1\) when \(t = 0\).
Using partial fractions, solve the differential equation, obtaining an expression for \(t\) in terms of \(x\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A curve is such that the gradient at a general point with coordinates \(( x , y )\) is proportional to \(\frac { y } { \sqrt { x + 1 } }\). The curve passes through the points with coordinates \(( 0,1 )\) and \(( 3 , e )\). By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).

7 \includegraphics[max width=\textwidth, alt={}, center]{1990cbac-d96f-4484-be4b-67dab35b3147-10_647_519_260_813} For the curve shown in the diagram, the normal to the curve at the point \(P\) with coordinates \(( x , y )\) meets the \(x\)-axis at \(N\). The point \(M\) is the foot of the perpendicular from \(P\) to the \(x\)-axis. The curve is such that for all values of \(x\) in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\), the area of triangle \(P M N\) is equal to \(\tan x\).

1. 1. Show that \(\frac { M N } { y } = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence show that \(x\) and \(y\) satisfy the differential equation \(\frac { 1 } { 2 } y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \tan x\).
2. Given that \(y = 1\) when \(x = 0\), solve this differential equation to find the equation of the curve, expressing \(y\) in terms of \(x\).

4 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x y } { 1 + x ^ { 2 } }$$ and \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining a simplified expression for \(y\) in terms of \(x\).

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { y - x } ,$$ and \(y = 0\) when \(x = 0\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. Find the value of \(y\) when \(x = 1\), giving your answer in the form \(a - \ln b\), where \(a\) and \(b\) are integers.

8 At time \(t\) days after the start of observations, the number of insects in a population is \(N\). The variation in the number of insects is modelled by a differential equation of the form \(\frac { \mathrm { d } N } { \mathrm {~d} t } = k N ^ { \frac { 3 } { 2 } } \cos 0.02 t\), where \(k\) is a constant and \(N\) is a continuous variable. It is given that when \(t = 0 , N = 100\).

1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
2. Given also that \(N = 625\) when \(t = 50\), find the value of \(k\).
3. Obtain an expression for \(N\) in terms of \(t\), and find the greatest value of \(N\) predicted by this model.

7 The variables \(x\) and \(y\) satisfy the differential equation $$\cos 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 4 \tan 2 x } { \sin ^ { 2 } 3 y }$$ where \(0 \leqslant x < \frac { 1 } { 4 } \pi\). It is given that \(y = 0\) when \(x = \frac { 1 } { 6 } \pi\).
Solve the differential equation to obtain the value of \(x\) when \(y = \frac { 1 } { 6 } \pi\). Give your answer correct to 3 decimal places.

8

1. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 + 9 y ^ { 2 } } { \mathrm { e } ^ { 2 x + 1 } } .$$ It is given that \(y = 0\) when \(x = 1\).
   Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. State what happens to the value of \(y\) as \(x\) tends to infinity. Give your answer in an exact form.

8 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } + 4 } { x ( y + 4 ) }$$ for \(x > 0\). It is given that \(x = 4\) when \(y = 2 \sqrt { 3 }\).
Solve the differential equation to obtain the value of \(x\) when \(y = 2\).

11 In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time \(t\) after the first plant is infected there are \(x\) infected plants. The rate of change of \(x\) is proportional to the product of the number of plants infected and the number of plants that are not yet infected. The variables \(x\) and \(t\) are treated as continuous, and it is given that \(\frac { \mathrm { dx } } { \mathrm { dt } } = 0.2\) and \(x = 1\) when \(t = 0\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$1495 \frac { \mathrm { dx } } { \mathrm { dt } } = x ( 300 - x )$$
2. Using partial fractions, solve the differential equation and obtain an expression for \(t\) in terms of a single logarithm involving \(x\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

10

1. By writing \(y = \sec ^ { 3 } \theta\) as \(\frac { 1 } { \cos ^ { 3 } \theta }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = 3 \sin \theta \sec ^ { 4 } \theta\).
2. The variables \(x\) and \(\theta\) satisfy the differential equation $$\left( x ^ { 2 } + 9 \right) \sin \theta \frac { d \theta } { d x } = ( x + 3 ) \cos ^ { 4 } \theta$$ It is given that \(x = 3\) when \(\theta = \frac { 1 } { 3 } \pi\).
   Solve the differential equation to find the value of \(\cos \theta\) when \(x = 0\). Give your answer correct to 3 significant figures.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The equation of a curve is \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 5\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + y ^ { 2 } } { y ^ { 2 } - 2 x y }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(y\)-axis.

9 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) ( 3 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\).
Solve the differential equation and find the exact value of \(y\) when \(x = 3\), giving your answer in a simplified form.