Question

If ax²+2hxy+by²=1 prove that d²y/dx²=h²-ab/(hx+by)³

Answer 1

Answer:

ddx(ax2+2hxy+by2)2ax+2h(xdydx+y)product rule+2bydydxdydxd2ydx2d2ydx2d2ydx2d2ydx2d2ydx2d2ydx2d2ydx2=0=0=−ax+hyhx+by=−[(hx+by)ddx(ax+hy)−(ax+hy)ddx(hx+by)(hx+by)2]=−⎡⎣⎢(hx+by)(a+hdydx)−(ax+hy)(h+bdydx)(hx+by)2⎤⎦⎥=−⎡⎣(h2x−abx)dydx+aby−h2y(hx+by)2⎤⎦=−(h2−ab)⎡⎣xdydx−y(hx+by)2⎤⎦=−(h2−ab)⎡⎣⎢−x(ax+hyhx+by)−y(hx+by)2⎤⎦⎥=(h2−ab)[ax2+2hxy+by2(hx+by)3]=(h2−ab)[1(hx+by)3]