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  • The equation of the curve satisfying the differential equation <img alt="y_{2} (x^{2} + 1) = 2xy_{1}" src="https://learn.careers360.com/latex-image/?y_%7B2%7D%20%28x%5E%7B2%7D%20&plus;%201%29%2

The equation of the curve satisfying the differential equation y_{2} (x^{2} + 1) = 2xy_{1} passing through the point (0, 1) and having slope of tangnet at x = 0 as 3, is

Option: 1

y=x^{2}+3x+2
 


Option: 2

y^{2}=x^{2}+3x+1

 


Option: 3

y=x^{3}+3x+1


Option: 4

none of these


Answers (1)

best_answer

 

Variable Separation Method -

integrating, we get

\int f\left ( x \right )dx+\int \phi \left ( y \right )dy =c

-

 

 

\frac{d^{2}y}{dx^{2}}(x^{2}+1)=2x\frac{dy}{dx}

\Rightarrow \int \frac{\frac{d^{2}y}{dx^{2}}}{\frac{dy}{dx}}dx=\int \frac{2x}{x^{2}+1}dx

\Rightarrow \ ln \left ( \frac{dy}{dx} \right )=\ ln(x^{2}+1)+ c

\\\Rightarrow \ ln \left ( \frac{dy}{dx} \right )-\ ln(x^{2}+1)+ c\\\Rightarrow \;\;\;\ln\left ( \frac{dy/dx}{x^2+1} \right )=c\\\Rightarrow \;\;\;\;\frac{1}{1+x^2}\cdot \frac{dy}{dx}=e^c\\at\;x=0,\;\;e^c=3\\\;\frac{1}{1+x^2}\cdot \frac{dy}{dx}=3\Rightarrow \frac{dy}{dx}=3(1+x^2)\\y=3x+x^3+c

 

\Rightarrow y=x^{3}+3x+1

Posted by

Rishabh

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