What is a particular solution to the differential equation dydx=xy with y(1)=2?

Explanation:this is separabley′=xyy y'=x∫yy′dx=∫xdx∫ddx(y22)dx=∫xdxy22=x22+Cy2=x2+Cy=±x2+Cy(1)=22=±1+Csuggest the + root solution. so2=1+C⇒C=3y=x2+3

Answered: "What is a particular solution to the differential..."

Adison Palmer

2022-04-18

What is a particular solution to the differential equation

\frac{d y}{d x} = \frac{x}{y}

with y(1)=2?

cloirdxti

Beginner2022-04-19Added 12 answers

Explanation:
this is separable

y^{'} = \frac{x}{y}

y y'=x

\int y y^{'} d x = \int x d x

\int \frac{d}{d x} (\frac{y^{2}}{2}) d x = \int x d x

\frac{y^{2}}{2} = \frac{x^{2}}{2} + C

y^{2} = x^{2} + C

y = \pm \sqrt{x^{2} + C}

y(1)=2

2 = \pm \sqrt{1 + C}

suggest the + root solution. so

2 = \sqrt{1 + C} \Rightarrow C = 3

y = \sqrt{x^{2} + 3}

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