What is a particular solution to the differential equation #dy/dx=-x/y# with #y(4)=3#?

Question

Elijah Abram · Accepted Answer

#y = pm sqrt(25 - x^2)#

equation is separable as follows

#y dy/dx = -x#

#int \ y dy/dx \ dx =int\ -x \ dx#

#int \ y \ dy = - int\ x \ dx#

#y^2/2 = - (x^2/2 - C)#

#y^2/2 = C - x^2/2#

#y^2 = C - x^2#

applying the IV

#9 = C - 4^2, implies C = 19#

#y^2 = 25 - x^2#

#y = pm sqrt(25 - x^2)#

Charles Anctil · Answer

undefined To find a particular solution to the differential equation $ \frac{dy}{dx} = -\frac{x}{y} $ with $ y(4) = 3 $, we can follow these steps:

1. Separate variables to get $ y \, dy = -x \, dx $.
2. Integrate both sides to obtain $ \int y \, dy = -\int x \, dx $.
3. Solve the integrals to get $ \frac{y^2}{2} = -\frac{x^2}{2} + C $, where $ C $ is the constant of integration.
4. Plug in the initial condition $ y(4) = 3 $ to find $ C $.
5. Substitute the value of $ C $ back into the equation to get the particular solution.

Let's solve step by step:

1. Separate variables: $ y \, dy = -x \, dx $.
2. Integrate both sides: $ \int y \, dy = -\int x \, dx $.
3. Solve the integrals: $ \frac{y^2}{2} = -\frac{x^2}{2} + C $, where $ C $ is the constant of integration.
4. Use the initial condition $ y(4) = 3 $ to find $ C $: $ \frac{3^2}{2} = -\frac{4^2}{2} + C $, which simplifies to $ \frac{9}{2} = -8 + C $. Solving for $ C $ gives $ C = \frac{25}{2} $.
5. Substitute $ C = \frac{25}{2} $ back into the equation: $ \frac{y^2}{2} = -\frac{x^2}{2} + \frac{25}{2} $.

Thus, the particular solution to the differential equation with the given initial condition is $ y^2 = -x^2 + 25 $.