A-LevelMathematicsDifferential equations9709/32 · Oct/Nov 2021 · Q77 Marks
The variables x and y satisfy the differential equation e²ˣ dy/dx = 4xy², and it is given that y = 1 when x = 0. Solve the differential equation, obtaining an expression for y in terms of x.
- The variables x and y satisfy the differential equation e²ˣ dy/dx = 4xy², and it is given that y = 1 when x = 0. Solve the differential equation, obtaining an expression for y in terms of x.[7]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.
- [B1] Separate variables correctly
- [B1] 1/y² dy = 4xe⁻²ˣ dx OE
- [M1] Commence the other integration and reach axe⁻²ˣ + b∫e⁻²ˣ dx
- [A1] Obtain -2xe⁻²ˣ + 2∫e⁻²ˣ dx or −2xe⁻²ˣ + ½ ∫e⁻²ˣ dx
- [A1] Complete integration and obtain -2xe⁻²ˣ - e⁻²ˣ
- [M1] Evaluate a constant or use x = 0 and y = 1 as limits in a solution containing terms of the form 1/y, qxe⁻²ˣ, re⁻²ˣ, or equivalent.
- [A1] Obtain y = e²ˣ / (2x+1), or equivalent expression for y
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