Find the particular solution of the differential equation
\(49 \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+14 \frac{\mathrm{~d} y}{\mathrm{~d} x}+y=49 x+735,\)
given that when \(x=0, y=0\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=0\).
The variables \(x\) and \(y\) are such that \(y=-1\) when \(x=0\) and
\(\left(x+\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^{3}=y^{2}+x\)
(i) Find the value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when \(x=0\).
(ii) Find also the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) when \(x=0\).
The curve \(C\) has equation
\(x^{3}+2 x y+8 y^{3}=-12\)
(a) Show that, at the point \((-2,-1)\) on \(C, \frac{\mathrm{~d} y}{\mathrm{~d} x}=-\frac{1}{2}\).
(b) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \((-2,-1)\).
The curve \(C\) has equation
\(4 y^{3}+(x+y)^{6}=109 .\)
(a) Show that, at the point \((-4,3)\) on \(C, \frac{\mathrm{~d} y}{\mathrm{~d} x}=\frac{1}{17}\).
(b) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \((-4,3)\).
The curve \(C\) has equation
\(4 y^{2}+4 \ln (x y)=1 .\)
(a) Show that, at the point \(\left(2, \frac{1}{2}\right)\) on \(C, \frac{\mathrm{~d} y}{\mathrm{~d} x}=-\frac{1}{6}\).
(b) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \(\left(2, \frac{1}{2}\right)\).
A curve has equation
\((x+1) y+y^{2}=2\)
(a) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{2}{3}\) at the point \((0,-2)\).
(b) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \((0,-2)\).
The curve \(C\) has equation
\(x y^{3}-4 x^{3} y=3 .\)
(a) Show that, at the point \((-1,1)\) on \(C, \frac{\mathrm{~d} y}{\mathrm{~d} x}=11\).
(b) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \((-1,1)\).
The curve \(C\) has equation
\[y^{2}+(x y+1)^{2}=5 .\]
(a) Show that, at the point \((1,1)\) on \(C, \frac{\mathrm{~d} y}{\mathrm{~d} x}=-\frac{2}{3}\).
(b) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \((1,1)\).
A curve \(C\) has equation \(\tan y=x\), for \(x\gt 0\).
(i) Use implicit differentiation to show that
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=-2 x\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^{2} .\)
(ii) Hence find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \(\left(1, \frac{1}{4} \pi\right)\) on \(C\).
The curve \(C\) has equation \(y=\frac{x^{2}-3 x+6}{1-x}\).
(i) Find the equations of the asymptotes of \(C\).
(ii) Find the coordinates of the turning points of \(C\).
(iii) Find the coordinates of any intersections with the coordinate axes.
(iv) Sketch \(C\).
It is given that \(x=t^{\frac{1}{2}}\), where \(x\gt 0\) and \(t\gt 0\), and \(y\) is a function of \(x\).
(i) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=2 t^{\frac{1}{2}} \frac{\mathrm{~d} y}{\mathrm{~d} t}\) and \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2 \frac{\mathrm{~d} y}{\mathrm{~d} t}+4 t \frac{\mathrm{~d}^{2} y}{\mathrm{~d} t^{2}}\).
(ii) Hence show that the differential equation
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-\left(8 x+\frac{1}{x}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}+12 x^{2} y=4 x^{2} \mathrm{e}^{-x^{2}}\)
reduces to the differential equation
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}}-4 \frac{\mathrm{~d} y}{\mathrm{~d} t}+3 y=\mathrm{e}^{-t} .\)
(iii) Find the general solution of \((*)\), giving \(y\) in terms of \(x\).
A curve \(C\) has equation \(x^{3}-3 x y+y^{2}=4\). Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \((0,2)\) of \(C\).
The curve \(C\) is defined parametrically by
\(x=2 \cos ^{3} t \quad \text { and } \quad y=2 \sin ^{3} t, \quad \text { for } 0\lt t\lt \frac{1}{2} \pi .\)
Show that, at the point with parameter \(t\),
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{1}{6} \sec ^{4} t \operatorname{cosec} t\)
The curve \(C\) has equation \(x^{2}+2 x y-4 y^{2}+20=0\). Show that if the tangent to \(C\) at the point \((x, y)\) is parallel to the \(x\)-axis then \(x+y=0\).
Hence find the coordinates of the stationary points on \(C\), and determine their nature.
A curve has equation \(x^{2}-6 x y+25 y^{2}=16\). Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=0\) at the point \((3,1)\).
By finding the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \((3,1)\), determine the nature of this turning point.
A curve \(C\) has equation \(x^{2}+4 x y-y^{2}+20=0\). Show that, at stationary points on \(C, x=-2 y\).
Find the coordinates of the stationary points on \(C\), and determine their nature by considering the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the stationary points.
The curve \(C\) has equation \(2 x^{3}+3 x^{2} y-3 y^{3}-16=0\).
(i) Find the coordinates of the point \(A\) on \(C\) at which \(\frac{\mathrm{d} y}{\mathrm{~d} x}=0\) and \(x \neq 0\).
(ii) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at \(A\).
Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{4}{3}\) at the point \(A(1,-2)\) on the curve with equation
\(y^{3}-3 x^{2} y+2=0,\)
and find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at \(A\).
Find the value of the constant \(\lambda\) such that \(\lambda x \mathrm{e}^{-x}\) is a particular integral of the differential equation
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+5 \frac{\mathrm{~d} y}{\mathrm{~d} x}+4 y=6 \mathrm{e}^{-x}\)
Find the solution of the differential equation for which \(y=2\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=3\) when \(x=0\).
The curve \(C\) has equation
\(y=\frac{p x^{2}+4 x+1}{x+1},\)
where \(p\) is a positive constant and \(p \neq 3\).
(i) Obtain the equations of the asymptotes of \(C\).
(ii) Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
(iii) For the case \(p=1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.