Question

Soles the differential equations using the substitution u=y’; u’=y”

y’y”=2

Answer 1

Answer with explanation:

The given differential equation

y'y''=2--------(1)

We have to apply the following substitution

u=y'

u'=y"

Applying these substitution in equation (1)

u u'=2

`u (du)/(dx)=2\\\\ u du=2 dx\\\\ \int u du=\int 2 dx\\\\(u^2)/(2)=2 x+K\\\\(y'^2)/(2)=2 x+K\\\\y'^2=4 x+2 K\\\\y'=(4 x+2 K)^{(1)/(2)}\\\\ dy=(4 x+2 K)^{(1)/(2)} d x\\\\\int dy=\int(4 x+2 K)^{(1)/(2)} d x\\\\y=\frac{(4 x+2 K)^{(3)/(2)}}{4 * (3)/(2)}+J\\\\y=\frac{(4 x+2 K)^{(3)/(2)}}{6}+J`

Where , J and K are constant of Integration.