What is the general solution of the differential y'= y/(x+1)



Answer to a math question What is the general solution of the differential y'= y/(x+1)

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Given differential equation is: \frac{dy}{dx} = \frac{y}{x+1} .

Step 1: Separate the variables:
\frac{dy}{y} = \frac{dx}{x+1}

Step 2: Integrate both sides:
\int\frac{1}{y} dy = \int\frac{1}{x+1} dx

Step 3: Solve the integrals:
ln|y| = ln|x+1| + C

Step 4: Eliminate the natural logarithms:
e^{ln|y|} = e^{ln|x+1| + C}

Step 5: Simplify the equation:
y = e^C \cdot |x + 1|

Step 6: Finally, rewrite the arbitrary constant as e^C = C_1 :
\boxed{y = C_1 \cdot |x + 1|} , where C_1 is an arbitrary constant.

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