Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/177

EXAMPLES

Change the independent variable from x to y in the four following equations:

1. $R={\frac {\left[1+\left({\frac {dy}{dx}}\right)^{2}\right]^{\frac {3}{2}}}{\frac {d^{2}y}{dx^{2}}}}$	Ans.	$R=-{\frac {\left[1+\left({\frac {dx}{dy}}\right)^{2}\right]^{\frac {3}{2}}}{\frac {d^{2}x}{dy^{2}}}}$
2. ${\frac {d^{2}y}{dx^{2}}}+2y\left({\frac {dy}{dx}}\right)^{2}=0$.		${\frac {d^{2}x}{dy^{2}}}-2y{\frac {dx}{dy}}=0$.
3. $x{\frac {d^{2}y}{dx^{2}}}+\left({\frac {dy}{dx}}\right)^{3}-{\frac {dy}{dx}}=0$		$x{\frac {d^{2}y}{dx^{2}}}-1+\left({\frac {dx}{dy}}\right)^{2}=0$.
4. $\left(3a{\frac {dy}{dx}}+2\right)\left({\frac {d^{2}y}{dx^{2}}}\right)^{2}=\left(a{\frac {dy}{dx}}+1\right){\frac {dy}{dx}}{\frac {d^{3}y}{dx^{3}}}$		$\left({\frac {d^{2}x}{dy^{2}}}\right)^{2}=\left({\frac {dx}{dy}}+a\right){\frac {d^{3}x}{dy^{3}}}$.

Change the dependent variable from y to z in the following equations:

5. $(1+y)^{2}\left({\frac {d^{3}y}{dx^{3}}}-2y\right)+\left({\frac {dy}{dx}}\right)^{2}=2\left(1+y\right){\frac {dy}{dx}}{\frac {d^{2}y}{dx^{2}}},y=z^{2}+2z$.
	Ans.	$(z+1){\frac {d^{3}x}{dx^{3}}}={\frac {dz}{dx}}{\frac {d^{2}z}{dx^{2}}}+z^{2}+2z$.
6. ${\frac {d^{2}y}{dx^{2}}}=1+{\frac {2(1+y)}{1+y^{2}}}\left({\frac {dy}{dx}}\right)^{2},y=\tan z$.	Ans.	${\frac {d^{2}z}{dx^{2}}}-2\left({\frac {dz}{dx}}\right)^{2}=\cos ^{2}z$.
$y^{2}{\frac {d^{3}y}{dx^{3}}}-\left(3y{\frac {dy}{dx}}+2xy^{2}\right){\frac {d^{2}y}{dx^{2}}}+\left\{2\left({\frac {dy}{dx}}\right)^{2}2xy{\frac {dy}{dx}}+3x^{2}y^{2}\right\}{\frac {dy}{dx}}+x^{3}y^{3}=0,y=e^{z}$.
	Ans.	${\frac {d^{3}z}{dx^{3}}}-2x{\frac {d^{2}z}{dx^{2}}}+3x^{2}{\frac {dz}{dx}}+x^{3}=0$.

Change the independent variable in the following eight equations:

8. ${\frac {d^{2}y}{dx^{2}}}-{\frac {x}{1-x^{2}}}{\frac {dy}{dx}}+{\frac {y}{1-x^{2}}}=0$,	$x=\cos t$.	Ans.	${\frac {d^{2}y}{dt^{2}}}+y=0$.
9. $(1-x^{2}){\frac {d^{2}y}{dx^{2}}}-x{\frac {dy}{dx}}=0$,	$x=\cos z$.		${\frac {d^{2}y}{dz^{2}}}=0$.
10. $(1-y^{2}){\frac {d^{2}u}{dy^{2}}}-y{\frac {du}{dy}}+a^{2}u=0$,	$y=\sin x$.		${\frac {d^{2}u}{dx^{2}}}+a^{2}u=0$.
11. $x^{2}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}+{\frac {a^{2}}{x^{2}}}y=0$,	$x={\frac {1}{z}}$.		${\frac {d^{2}y}{dz^{2}}}+a^{2}y=0$.
12. $x^{3}{\frac {d^{3}v}{dx^{3}}}+3x^{2}{\frac {d^{2}v}{dx^{2}}}+x{\frac {dv}{dx}}+v=0$,	$x=e^{t}$.		${\frac {d^{3}v}{dx^{3}}}+v=0$.
13. ${\frac {d^{2}y}{dx^{2}}}+{\frac {2x}{1+x^{2}}}{\frac {dy}{dx}}+{\frac {y}{(1+x^{2})^{2}}}=0$,	$x=\tan \theta$.		${\frac {d^{2}y}{d\theta ^{2}}}+y=0$.
14. ${\frac {d^{2}u}{ds^{2}}}+su{\frac {du}{ds}}+\sec ^{2}s=0$,	$s=\arctan t$.
Ans. $(1+t^{2}){\frac {d^{2}u}{dt^{2}}}+(2t+x\arctan t){\frac {du}{dt}}+1=0$.
15. $x^{4}{\frac {d^{2}y}{dx^{2}}}+a^{2}y=0$,	$x={\frac {1}{z}}$.	Ans.	${\frac {d^{2}y}{dz^{2}}}+{\frac {2}{z}}{\frac {dy}{dz}}+a^{2}y=0$.