Final answer to the problem
$y\ln\left(y\right)-y=-\frac{1}{2}x^2\ln\left(x\right)+\frac{1}{4}x^2+C_0$
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Step-by-step Solution
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1
Multiplying polynomials $\ln\left(y\right)$ and $dy-x\cdot dx$
$x\ln\left(xy\right)\cdot dx+\ln\left(y\right)\cdot dy-x\ln\left(y\right)\cdot dx=0$
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$x\ln\left(xy\right)\cdot dx+\ln\left(y\right)\cdot dy-x\ln\left(y\right)\cdot dx=0$
Learn how to solve problems step by step online. Solve the differential equation xln(xy)dx+ln(y)(dy-xdx)=0. Multiplying polynomials \ln\left(y\right) and dy-x\cdot dx. Group the terms of the equation. Factor the polynomial -x\ln\left(xy\right)\cdot dx+x\ln\left(y\right)\cdot dx by it's greatest common factor (GCF): x\cdot dx. Applying the product rule for logarithms: \log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right).
Final answer to the problem
$y\ln\left(y\right)-y=-\frac{1}{2}x^2\ln\left(x\right)+\frac{1}{4}x^2+C_0$