Let y=y(x) be the solution of the differential equation ( x)^1/2dy=( ^3x-( x)^3/2y)dx,0

Question

Let $y=y(x)$ be the solution of the differential equation $(	an x{)}^{1/2}dy=({\sec }^{3}x-(	an x{)}^{3/2}y)dx,0$

Accepted Answer

$\frac{dy}{dx}+y	an x=\frac{{\sec }^{3}x}{\sqrt{	an x}}$

$IF={e}^{\int 	an x\, dx}=\sec x$

$y\sec x=\int \frac{{\sec }^{4}x}{\sqrt{	an x}}\, dx$

$y\sec x=\int \frac{(1+{	an }^{2}x)}{\sqrt{	an x}}{\sec }^{2}x\, dx$

$	an x=t$

$y\sec x=2\sqrt{	an x}+\frac{2}{5}(	an x{)}^{5/2}+c$

$y(\frac{\pi }{4})=\frac{6\sqrt{2}}{5}$

$\frac{6\sqrt{2}}{5}	imes \sqrt{2}=2+\frac{2}{5}+c	o \frac{12}{5}=\frac{12}{5}+c$

$	o c=0$

$y(\frac{\pi }{3})=\frac{8}{5}\cdot {3}^{1/4}	o \alpha =2	imes {3}^{1/4}$

$	herefore {\alpha }^{4}=48$

Solutions · 1