Question

Solve the separable differential equation

10x-8ysqrt(x^2 +1) * dy/dx =0

Subject to the initial condition: .y(0)=9

y=??

Answer 1

Let us solve the differential equation

$10x-8y\sqrt{x^2 +1}\frac{ dy}{dx }=0,$

which is equivalent to

$\frac{10x dx}{ \sqrt{x^2 +1}}=8ydy.$

It follows that

$\int\frac{10x dx}{ \sqrt{x^2 +1}}=\int8ydy.$

Therefore, the general solution is

$10\sqrt{x^2+1}=4y^2+C.$

Since $y(0)=9,$ we get $10=4\cdot 81+C.$

Therefore, $C=-314.$

Consequently, the particular solution is

$10\sqrt{x^2+1}=4y^2-314.$