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

equation is separable as follows

applying the IV

By signing up, you agree to our Terms of Service and Privacy Policy

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

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

Let's solve step by step:

- Separate variables: ( y , dy = -x , dx ).
- Integrate both sides: ( \int y , dy = -\int x , dx ).
- Solve the integrals: ( \frac{y^2}{2} = -\frac{x^2}{2} + C ), where ( C ) is the constant of integration.
- 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} ).
- 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 ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the volume of the solid obtained by rotating the region bounded by the curves #y=x^2+1# and #y=-x+3# rotated around the x-axis?
- How do you find the volume bounded by #y=sqrt(x + 1)#, x = 0, x = 3, and y = 0 revolved about the x-axis?
- What is the general solution of the differential equation? : # dy/dx = x+2y #
- What is the arc length of #f(x)=lnx # in the interval #[1,5]#?
- What are separable differential equations?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7