Question

Q. 2# (x+1)frac{dy}{dx}=x(y^{2}+1)

Second order linear equations
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asked 2021-02-08
Q. 2# \((x+1)\frac{dy}{dx}=x(y^{2}+1)\)

Answers (1)

2021-02-09
Here the Differential equations is given by
\((x+1)\frac{dy}{dx}=x(y^{2}+1)\)
\(\rightarrow\frac{dy}{y^{2}+1}= \frac{xdx}{x+1}\)
\(\rightarrow\int\frac{dy}{y^{2}+1}=\int\frac{xdx}{x+1}\)
\(\rightarrow \tan^{−1}(y)=\int\frac{x+1−1dx}{(x+1)+c}\)
\(\rightarrow \tan^{−1}(y)=\int\frac{(x+1−1)dx)}{x+1+c}\)
\(\rightarrow \tan^{−1}(y)=\int dx−\int\frac{dx}{(x+1)+c}\)
\(\rightarrow \tan^{−1}(y)=x−\ln∣x+1∣+c\)
\(\rightarrow y=tan(x−\ln∣x+1∣+c),\)
where cc is a arbitrary constant.
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