CAIE A-Level Further Math A2 2.6.5 Differential Substitutions QuestionsPractise solving differential equations using substitutions that simplify the dependent expression with Further Mathematics Paper 2 questions and mark schemes.Syllabus2028–2030CourseFurther Mathematics 9231LevelA2
CAIE A-Level Further Math A2 2.6.5 Differential Substitutions Questions question 1[Maximum number: 4]It is given that x=t3yx=t^{3} yx=t3y andt3 d2y dt2+(4t3+6t2)dy dt+(13t3+12t2+6t)y=61e12t.t^{3} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} t^{2}}+\left(4 t^{3}+6 t^{2}\right) \frac{\mathrm{d} y}{\mathrm{~d} t}+\left(13 t^{3}+12 t^{2}+6 t\right) y=61 \mathrm{e}^{\frac{1}{2} t} .t3 dt2 d2y+(4t3+6t2) dtdy+(13t3+12t2+6t)y=61e21t.Show thatd2x dt2+4 dx dt+13x=6e12t\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{~d} x}{\mathrm{~d} t}+13 x=6 \mathrm{e}^{\frac{1}{2} t} dt2d2x+4 dt dx+13x=6e21tShow Answerdx dt=t3 dy dt+3t2y\frac{\mathrm{d} x}{\mathrm{~d} t}=t^{3} \frac{\mathrm{~d} y}{\mathrm{~d} t}+3 t^{2} y dtdx=t3 dt dy+3t2yd2x dt2=t3 d2y dt2+6t2 dy dt+6ty\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}=t^{3} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} t^{2}}+6 t^{2} \frac{\mathrm{~d} y}{\mathrm{~d} t}+6 t y dt2d2x=t3 dt2 d2y+6t2 dt dy+6tyd2x dt2+4 dx dt+13x=t3 d2y dt2+6t2 dy dt+6ty+4t3 dy dt+12t2y+13t3y=61e12t\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{~d} x}{\mathrm{~d} t}+13 x=t^{3} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} t^{2}}+6 t^{2} \frac{\mathrm{~d} y}{\mathrm{~d} t}+6 t y+4 t^{3} \frac{\mathrm{~d} y}{\mathrm{~d} t}+12 t^{2} y+13 t^{3} y=61 \mathrm{e}^{\frac{1}{2} t} dt2d2x+4 dt dx+13x=t3 dt2 d2y+6t2 dt dy+6ty+4t3 dt dy+12t2y+13t3y=61e21t7a 4Add to Test