Zadanie 9, Paper 1, maj 2026, TZ2, Matematyka AA IB HL

Separate: $\int y/(4+y^2)\,dy=\int1/x\,dx$, giving $\frac12\ln(4+y^2)=\ln|x|+C$. Thus 4+y²=Ax². At (3,√5), 9=9A, so A=1 and y²=x²−4. The specified branch is y=√(x²−4), x>2; it contains the initial point and satisfies the differential equation.

M1: separate variables or make an equivalent useful substitution. A1: left integral ½ln(4+y²). A1: right integral ln|x|+C. M1: apply the initial condition. A1: correct constant A=1, or C=0. A1: correct logarithmic/exponentiated relation 4+y²=x². A1: y²=x²−4. The stated branch makes the problem regular; a branch/domain statement is explanatory and not an extra required mark. Condone absence of the integration constant at the initial integration stage and absence of logarithmic modulus signs on this positive-x branch. Alternative v=y/x: M1 form v dv/dx=4/x^3; A1 for each integrated side v^2/2=-2/x^2+C; A1 substitute back v=y/x; M1 use the initial condition; A1 C=1/2; A1 y^2=x^2-4. Equivalent order of these steps is accepted.