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  • Solve: 2(y+3)-xy\frac{dy}{dx}=0 given that y (1) = –2

Solve: 2(y+3)-xy\frac{dy}{dx}=0 given that y (1) = –2

 

Answers (1)

Given:

2\left ( y+3 \right )-xy\frac{dy}{dx}=0

To Find: Solution of the differential equation

2\frac{dx}{x}=\frac{ydy}{y+3}\\ 2\frac{dx}{x}=\frac{\left [ \left ( y+3 \right ) -3\right ]dy}{y+3}\\ 2\frac{dx}{x}=dy-\frac{3dy}{y+3}

Integrating on both sides

\int 2\frac{dx}{x}=\int dy-\int \frac{3dy}{y+3}\\ 2\ln x=y-3\ln(y+3)+c\\ Formula: \int \frac{dx}{x}=\ln x

Substitute (-2,1) to find value of c
\\0=-2+c \\c=2 \\2 \ln x =y-3 \ln(y+3)+2 \\ 2 \ln x+3\ln(y+3)=y+2 \\2 \ln x+3\ln(y+3)=y+2 \\ \ln x^{2}+ \ln(y+3)^{3}=y+2 \\x^{2}(y+3)^{3}=e^{y+2}

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infoexpert24

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