Question

Solve the differential equation:
\[(1-x^2)\frac{\text d y}{\text dx}+xy=3x\]

Answer 1

i see linear?

Answer 2

\[(1-x^2)\frac{\text d y}{\text dx}+xy=3x\]\[\frac{\text d y}{\text dx}+\frac{x}{1-x^2}y=\frac{3x}{1-x^2}\]

\[R(x)=e^{\int \frac{x}{1-x^2}\text dx }=e^{\frac{-1}{2}\int \frac{-2x}{1-x^2}\text dx}=e^{\frac{-1}2\ln |1-x^2|}=\frac 1{\sqrt{1-x^2}}\]
\[\frac{\text d }{\text dx}\left(\frac 1{\sqrt{1-x^2}}\cdot y\right)=\frac 1{\sqrt{1-x^2}}\cdot \frac{3x}{1-x^2}\]\[\frac{\text d }{\text dx}\left(\frac y{\sqrt{1-x^2}}\right)=\frac{3x}{(1-x^2)^{3/2}}\]\[\frac {y}{\sqrt{1-x^2}}=\int \frac{3x}{(1-x^2)^{3/2}}\text dx\]

Answer 3

sounds right..

Answer 4

\[\int \frac{3x}{(1-x^2)^{3/2}} dx\] that'sintegrable by u-sub right?

Answer 5

\[\frac {y}{\sqrt{1-x^2}}=\int \frac{3x}{(1-x^2)^{3/2}}\text dx\]
\[\frac {y}{\sqrt{1-x^2}}=\frac{3}{\sqrt{1-x^2}}+c_1\]

Answer 6

yes i got the same answer as u did

Answer 7

\[y=3+c \sqrt{1-x^2}\]

Answer 8

yup sounds right to me ^_^