DP Further Mathematics HL Questionbank

• Syllabus

5.5

Sub sections and their related questions

First-order differential equations.

• 11M.1.hl.TZ0.2b: Find the solution to the differential...
• 10M.1.hl.TZ0.5: Given that \(\frac{{{\rm{d}}x}}{{{\rm{d}}y}} + 2y\tan x = \sin x\) , and \(y = 0\) when...
• 09M.2.hl.TZ0.4B: Solve the differential equation \((u + 3{v^3})\frac{{{\rm{d}}v}}{{{\rm{d}}u}} = 2v\) , giving...
• 13M.1.hl.TZ0.4a: Find the general solution of the differential equation...
• 08M.1.hl.TZ0.5: Solve the following differential...
• 07M.1.hl.TZ0.5: Solve the differential equation \(x\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + 2y = \sqrt {1 + {x^2}}...
• 12M.2.hl.TZ0.3b: Consider the differential equation...
• SPNone.2.hl.TZ0.5a: Use Euler’s method with a step length of \(0.1\) to find an approximate value for \(y\) when...
• SPNone.2.hl.TZ0.5c: (i)     Show that \(\sec x + \tan x\) is an integrating factor for solving this differential...
• 14M.1.hl.TZ0.2: Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^3} - {x^3}\) for...
• 14M.2.hl.TZ0.5: Consider the differential equation...
• 15M.1.hl.TZ0.8b: Hence solve the differential equation...
• 16M.2.hl.TZ0.4a: Find the general solution of the differential equation, expressing your answer in the form...
• 16M.2.hl.TZ0.4b: (i)     Hence find the particular solution passing through the points...
• 16M.2.hl.TZ0.4c: (i)     Write down the particular solution passing through the points \((1,{\text{ }} \pm...
• 16M.2.hl.TZ0.4d: (i)     Find the general solution of the differential equation...
• 18M.1.hl.TZ0.11a: Show that...
• 18M.1.hl.TZ0.11b: Show that the differential...
• 18M.1.hl.TZ0.11c: Hence show that the solution to the differential...
• 18M.2.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of \(y\) when \(x = 0.4\).
• 18M.2.hl.TZ0.2b.i: Show...
• 18M.2.hl.TZ0.2b.ii: Show that...
• 18M.2.hl.TZ0.4a.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2\).
• 18M.2.hl.TZ0.4a.ii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + 1\).
• 18M.2.hl.TZ0.4a.iii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - 1\).
• 18M.2.hl.TZ0.4b.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \) constant.
• 18M.2.hl.TZ0.4c: The slope field for the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + y\)...
• 18M.2.hl.TZ0.4d: Given that all the isoclines from a slope field of a differential equation are straight lines...

Geometric interpretation using slope fields, including identification of isoclines.

• 18M.1.hl.TZ0.11a: Show that...
• 18M.1.hl.TZ0.11b: Show that the differential...
• 18M.1.hl.TZ0.11c: Hence show that the solution to the differential...
• 18M.2.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of \(y\) when \(x = 0.4\).
• 18M.2.hl.TZ0.2b.i: Show...
• 18M.2.hl.TZ0.2b.ii: Show that...
• 18M.2.hl.TZ0.4a.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2\).
• 18M.2.hl.TZ0.4a.ii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + 1\).
• 18M.2.hl.TZ0.4a.iii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - 1\).
• 18M.2.hl.TZ0.4b.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \) constant.
• 18M.2.hl.TZ0.4c: The slope field for the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + y\)...
• 18M.2.hl.TZ0.4d: Given that all the isoclines from a slope field of a differential equation are straight lines...

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

• 14M.1.hl.TZ0.2: Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^3} - {x^3}\) for...
• 15M.2.hl.TZ0.1a: Using Euler’s method with increments of \(0.2\), find an approximate value for \(y\) when \(x = 1\).
• 15M.2.hl.TZ0.1b: Explain how Euler’s method could be improved to provide a better approximation.
• 18M.1.hl.TZ0.11a: Show that...
• 18M.1.hl.TZ0.11b: Show that the differential...
• 18M.1.hl.TZ0.11c: Hence show that the solution to the differential...
• 18M.2.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of \(y\) when \(x = 0.4\).
• 18M.2.hl.TZ0.2b.i: Show...
• 18M.2.hl.TZ0.2b.ii: Show that...
• 18M.2.hl.TZ0.4a.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2\).
• 18M.2.hl.TZ0.4a.ii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + 1\).
• 18M.2.hl.TZ0.4a.iii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - 1\).
• 18M.2.hl.TZ0.4b.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \) constant.
• 18M.2.hl.TZ0.4c: The slope field for the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + y\)...
• 18M.2.hl.TZ0.4d: Given that all the isoclines from a slope field of a differential equation are straight lines...

Variables separable.

• 15M.1.hl.TZ0.8b: Hence solve the differential equation...
• 18M.1.hl.TZ0.11a: Show that...
• 18M.1.hl.TZ0.11b: Show that the differential...
• 18M.1.hl.TZ0.11c: Hence show that the solution to the differential...
• 18M.2.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of \(y\) when \(x = 0.4\).
• 18M.2.hl.TZ0.2b.i: Show...
• 18M.2.hl.TZ0.2b.ii: Show that...
• 18M.2.hl.TZ0.4a.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2\).
• 18M.2.hl.TZ0.4a.ii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + 1\).
• 18M.2.hl.TZ0.4a.iii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - 1\).
• 18M.2.hl.TZ0.4b.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \) constant.
• 18M.2.hl.TZ0.4c: The slope field for the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + y\)...
• 18M.2.hl.TZ0.4d: Given that all the isoclines from a slope field of a differential equation are straight lines...

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

• 18M.1.hl.TZ0.11a: Show that...
• 18M.1.hl.TZ0.11b: Show that the differential...
• 18M.1.hl.TZ0.11c: Hence show that the solution to the differential...
• 18M.2.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of \(y\) when \(x = 0.4\).
• 18M.2.hl.TZ0.2b.i: Show...
• 18M.2.hl.TZ0.2b.ii: Show that...
• 18M.2.hl.TZ0.4a.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2\).
• 18M.2.hl.TZ0.4a.ii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + 1\).
• 18M.2.hl.TZ0.4a.iii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - 1\).
• 18M.2.hl.TZ0.4b.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \) constant.
• 18M.2.hl.TZ0.4c: The slope field for the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + y\)...
• 18M.2.hl.TZ0.4d: Given that all the isoclines from a slope field of a differential equation are straight lines...

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

• 15M.2.hl.TZ0.1c: Solve the differential equation to find an exact value for \(y\) when \(x = 1\).
• 18M.1.hl.TZ0.11a: Show that...
• 18M.1.hl.TZ0.11b: Show that the differential...
• 18M.1.hl.TZ0.11c: Hence show that the solution to the differential...
• 18M.2.hl.TZ0.2a: Use Euler’s method with step length 0.1 to find an approximate value of \(y\) when \(x = 0.4\).
• 18M.2.hl.TZ0.2b.i: Show...
• 18M.2.hl.TZ0.2b.ii: Show that...
• 18M.2.hl.TZ0.4a.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2\).
• 18M.2.hl.TZ0.4a.ii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + 1\).
• 18M.2.hl.TZ0.4a.iii: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x - 1\).
• 18M.2.hl.TZ0.4b.i: \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \) constant.
• 18M.2.hl.TZ0.4c: The slope field for the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = x + y\)...
• 18M.2.hl.TZ0.4d: Given that all the isoclines from a slope field of a differential equation are straight lines...