DP Mathematics HL Questionbank

• Syllabus

9.5

Sub sections and their related questions

First-order differential equations.

• 12M.2.hl.TZ2.12a: Find an expression for v in terms of t .
• 12M.3ca.hl.TZ0.2a: Use Euler’s method, with a step length of 0.1, to find an approximate value of y when x = 0.5.
• 12M.3ca.hl.TZ0.2c: (i)     Solve the differential equation. (ii)     Find the value of a for which...
• 12M.3ca.hl.TZ0.3: Find the general solution of the differential equation...
• 12N.3ca.hl.TZ0.1a: Solve this differential equation by separating the variables, giving your answer in the form y =...
• 12N.3ca.hl.TZ0.1b: Solve the same differential equation by using the standard homogeneous substitution y = vx .
• 12N.3ca.hl.TZ0.1c: Solve the same differential equation by the use of an integrating factor.
• 12N.3ca.hl.TZ0.1d: If y = 20 when x = 2 , find y when x = 5 .
• 12N.3ca.hl.TZ0.2a: Use Euler’s method to find an approximation for the value of c , using a step length of h = 0.1 ....
• 12N.3ca.hl.TZ0.2b: You are told that if Euler’s method is used with h = 0.05 then \(c \simeq 2.7921\) , if it is...
• 12N.3ca.hl.TZ0.2c: Draw, by eye, the straight line that best fits these four points, using a ruler.
• 12N.3ca.hl.TZ0.2d: Use your graph to give the best possible estimate for c , giving your answer to three decimal...
• 08N.3ca.hl.TZ0.4: (a)     Show that the solution of the differential...
• 08M.1.hl.TZ1.13: A gourmet chef is renowned for her spherical shaped soufflé. Once it is put in the oven, its...
• 08N.2.hl.TZ0.9: The population of mosquitoes in a specific area around a lake is controlled by pesticide. The...
• 11M.2.hl.TZ2.13B: (a)     Using integration by parts, show that...
• 11M.3ca.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of y when x = 0.4.
• 11M.3ca.hl.TZ0.2b: Write down, giving a reason, whether your approximate value for y is greater than or less than...
• 09M.3ca.hl.TZ0.4: Consider the differential equation...
• 09M.1.hl.TZ1.13Part B: Let f be a function with domain \(\mathbb{R}\) that satisfies the...
• 09N.1.hl.TZ0.8: A certain population can be modelled by the differential equation...
• SPNone.3ca.hl.TZ0.2a: Show that this is a homogeneous differential equation.
• SPNone.3ca.hl.TZ0.2b: Find the general solution, giving your answer in the form \(y = f(x)\) .
• SPNone.3ca.hl.TZ0.3a: By finding the values of successive derivatives when x = 0 , find the Maclaurin series for y as...
• SPNone.3ca.hl.TZ0.3b: (i)     Differentiate the function \({{\text{e}}^x}(\sin x + \cos x)\) and hence show...
• 10M.3ca.hl.TZ0.1: Given that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} - 2{y^2} = {{\text{e}}^x}\) and y = 1 when x =...
• 10N.1.hl.TZ0.8: Find y in terms of x, given that...
• 13M.3ca.hl.TZ0.2a: Use Euler’s method with a step length of 0.1 to find an approximation to the value of y when x =...
• 13M.3ca.hl.TZ0.2b: (i)     Show that the integrating factor for solving the differential equation is \(\sec...
• 13M.2.hl.TZ2.10: The acceleration of a car is \(\frac{1}{{40}}(60 - v){\text{ m}}{{\text{s}}^{ - 2}}\), when its...
• 13M.2.hl.TZ2.12a: (i)     Show that the function \(y = \cos x + \sin x\) satisfies the differential equation. (ii)...
• 13M.2.hl.TZ2.12b: A different solution of the differential equation, satisfying y = 2 when \(x = \frac{\pi }{4}\),...
• 11N.1.hl.TZ0.13b: Find \(f(x)\).
• 11N.1.hl.TZ0.13c: Determine the largest possible domain of f.
• 11N.1.hl.TZ0.13d: Show that the equation \(f(x) = f'(x)\) has no solution.
• 11N.3ca.hl.TZ0.6: The real and imaginary parts of a complex number \(x + {\text{i}}y\) are related by the...
• 11M.2.hl.TZ1.14c: If the glass is filled completely, how long will it take for all the water to evaporate?
• 09M.2.hl.TZ1.8: (a)     Solve the differential equation...
• 09M.2.hl.TZ2.6: The acceleration in ms−2 of a particle moving in a straight line at time \(t\) seconds,...
• 14M.3ca.hl.TZ0.2b: Consider the differential...
• 13N.3ca.hl.TZ0.3: Consider the differential equation...
• 15M.3ca.hl.TZ0.2a: Show that \(y = \frac{1}{x}\int {f(x){\text{d}}x} \) is a solution of the differential...
• 15M.3ca.hl.TZ0.2b: Hence solve...
• 15N.3ca.hl.TZ0.5a: Show that the tangent to the curve \(y = f(x)\) at the point \((1,{\text{ }}0)\) is normal to the...
• 16N.3ca.hl.TZ0.1a: Show that \(1 + {x^2}\) is an integrating factor for this differential equation.
• 16N.3ca.hl.TZ0.1b: Hence solve this differential equation. Give the answer in the form \(y = f(x)\).
• 18M.3ca.hl.TZ0.5a: Solve the differential equation given that \(y =  - 1\) when \(x = 1\). Give your answer in the...
• 18M.3ca.hl.TZ0.5b.i: Show that the \(x\)-coordinate(s) of the points on the curve \(y = f\left( x \right)\) where...
• 18M.3ca.hl.TZ0.5b.ii: Deduce the set of values for \(p\) such that there are two points on the curve...

Geometric interpretation using slope fields, including identification of isoclines.

• 12M.2.hl.TZ2.12a: Find an expression for v in terms of t .
• 12M.3ca.hl.TZ0.2a: Use Euler’s method, with a step length of 0.1, to find an approximate value of y when x = 0.5.
• 12M.3ca.hl.TZ0.2c: (i)     Solve the differential equation. (ii)     Find the value of a for which...
• 12M.3ca.hl.TZ0.3: Find the general solution of the differential equation...
• 11M.3ca.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of y when x = 0.4.
• 11M.3ca.hl.TZ0.2b: Write down, giving a reason, whether your approximate value for y is greater than or less than...
• 14N.3ca.hl.TZ0.3a: Sketch, on one diagram, the four isoclines corresponding to \(f(x,{\text{ }}y) = k\) where \(k\)...
• 14N.3ca.hl.TZ0.3b: A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation...
• 14N.3ca.hl.TZ0.3c: A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation...
• 15N.3ca.hl.TZ0.5a: Show that the tangent to the curve \(y = f(x)\) at the point \((1,{\text{ }}0)\) is normal to the...
• 15N.3ca.hl.TZ0.5d: Explain why \(y = f(x)\) cannot cross the isocline \(x - {y^2} = 0\), for \(x > 1\).
• 15N.3ca.hl.TZ0.5e: (i)     Sketch the isoclines \(x - {y^2} =  - 2,{\text{ }}0,{\text{ }}1\). (ii)     On the same...
• 18M.3ca.hl.TZ0.5a: Solve the differential equation given that \(y =  - 1\) when \(x = 1\). Give your answer in the...
• 18M.3ca.hl.TZ0.5b.i: Show that the \(x\)-coordinate(s) of the points on the curve \(y = f\left( x \right)\) where...
• 18M.3ca.hl.TZ0.5b.ii: Deduce the set of values for \(p\) such that there are two points on the curve...

Numerical solution of \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {x,y} \right)\) using Euler’s method.

• 12M.2.hl.TZ2.12a: Find an expression for v in terms of t .
• 12M.3ca.hl.TZ0.2a: Use Euler’s method, with a step length of 0.1, to find an approximate value of y when x = 0.5.
• 12M.3ca.hl.TZ0.2c: (i)     Solve the differential equation. (ii)     Find the value of a for which...
• 12M.3ca.hl.TZ0.3: Find the general solution of the differential equation...
• 08M.3ca.hl.TZ2.3: (a)     (i)     Use Euler’s method to get an approximate value of y when x = 1.3 , taking steps...
• 11M.3ca.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of y when x = 0.4.
• 11M.3ca.hl.TZ0.2b: Write down, giving a reason, whether your approximate value for y is greater than or less than...
• 09M.3ca.hl.TZ0.4: Consider the differential equation...
• 10M.3ca.hl.TZ0.1: Given that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} - 2{y^2} = {{\text{e}}^x}\) and y = 1 when x =...
• 14N.3ca.hl.TZ0.3d: A curve, \(C\), passes through the point \((0,1)\) and satisfies the differential equation...
• 15N.3ca.hl.TZ0.5c: Use Euler’s method with steps of \(0.2\) to estimate \(f(2)\) to \(5\) decimal places.
• 18M.3ca.hl.TZ0.5a: Solve the differential equation given that \(y =  - 1\) when \(x = 1\). Give your answer in the...
• 18M.3ca.hl.TZ0.5b.i: Show that the \(x\)-coordinate(s) of the points on the curve \(y = f\left( x \right)\) where...
• 18M.3ca.hl.TZ0.5b.ii: Deduce the set of values for \(p\) such that there are two points on the curve...

Variables separable.

• 12M.2.hl.TZ2.12a: Find an expression for v in terms of t .
• 12M.3ca.hl.TZ0.2c: (i)     Solve the differential equation. (ii)     Find the value of a for which...
• 12M.3ca.hl.TZ0.3: Find the general solution of the differential equation...
• 08N.3ca.hl.TZ0.4: (a)     Show that the solution of the differential...
• 08M.1.hl.TZ1.13: A gourmet chef is renowned for her spherical shaped soufflé. Once it is put in the oven, its...
• 08N.2.hl.TZ0.9: The population of mosquitoes in a specific area around a lake is controlled by pesticide. The...
• 09M.1.hl.TZ1.13Part B: Let f be a function with domain \(\mathbb{R}\) that satisfies the...
• 09N.1.hl.TZ0.8: A certain population can be modelled by the differential equation...
• 10N.1.hl.TZ0.8: Find y in terms of x, given that...
• 09M.2.hl.TZ1.8: (a)     Solve the differential equation...
• 09M.2.hl.TZ2.6: The acceleration in ms−2 of a particle moving in a straight line at time \(t\) seconds,...
• 18M.3ca.hl.TZ0.5a: Solve the differential equation given that \(y =  - 1\) when \(x = 1\). Give your answer in the...
• 18M.3ca.hl.TZ0.5b.i: Show that the \(x\)-coordinate(s) of the points on the curve \(y = f\left( x \right)\) where...
• 18M.3ca.hl.TZ0.5b.ii: Deduce the set of values for \(p\) such that there are two points on the curve...

Homogeneous differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {\frac{y}{x}} \right)\) using the substitution \(y = vx\) .

• 12M.3ca.hl.TZ0.3: Find the general solution of the differential equation...
• 11M.3ca.hl.TZ0.3: Solve the differential...
• 09M.3ca.hl.TZ0.4: Consider the differential equation...
• 09N.3ca.hl.TZ0.1: Solve the differential...
• 10M.3ca.hl.TZ0.3: Solve the differential...
• 18M.3ca.hl.TZ0.5a: Solve the differential equation given that \(y =  - 1\) when \(x = 1\). Give your answer in the...
• 18M.3ca.hl.TZ0.5b.i: Show that the \(x\)-coordinate(s) of the points on the curve \(y = f\left( x \right)\) where...
• 18M.3ca.hl.TZ0.5b.ii: Deduce the set of values for \(p\) such that there are two points on the curve...

Solution of \(y' + P\left( x \right)y = Q\left( x \right)\), using the integrating factor.

• 12M.3ca.hl.TZ0.3: Find the general solution of the differential equation...
• 08M.3ca.hl.TZ1.3: (a)     Find an integrating factor for this differential equation. (b)     Solve the...
• 08M.3ca.hl.TZ2.3: (a)     (i)     Use Euler’s method to get an approximate value of y when x = 1.3 , taking steps...
• 08N.3ca.hl.TZ0.1: (a)     Show that the solution of the homogeneous differential...
• 09M.3ca.hl.TZ0.2: The variables x and y are related by \(\frac{{{\text{d}}y}}{{{\text{d}}x}} - y\tan x = \cos x\)...
• 10N.3ca.hl.TZ0.4: Solve the differential...
• 14N.3ca.hl.TZ0.2a: Use an integrating factor to show that the general solution for...
• 14N.3ca.hl.TZ0.2b: Given that \(w(t)\) is continuous, find the value of \(c\).
• 14N.3ca.hl.TZ0.2c: Write down (i)     the weight of the dog when bought from the pet shop; (ii)     an upper bound...
• 15M.3ca.hl.TZ0.2b: Hence solve...
• 15N.3ca.hl.TZ0.5b: Find \(g(x)\).
• 18M.3ca.hl.TZ0.5a: Solve the differential equation given that \(y =  - 1\) when \(x = 1\). Give your answer in the...
• 18M.3ca.hl.TZ0.5b.i: Show that the \(x\)-coordinate(s) of the points on the curve \(y = f\left( x \right)\) where...
• 18M.3ca.hl.TZ0.5b.ii: Deduce the set of values for \(p\) such that there are two points on the curve...