(x^2+1)dy/dx+3x(y-1)=0

Jasonbradley

3 min read Jun 17, 2024

Solving the Differential Equation: (x^2+1)dy/dx + 3x(y-1) = 0

This article will guide you through solving the given first-order differential equation:

(x^2+1)dy/dx + 3x(y-1) = 0

We will use the method of separation of variables to find the general solution.

1. Rearranging the Equation

First, we need to manipulate the equation to separate the variables x and y.

Divide both sides by (x^2+1):
```
dy/dx + (3x(y-1))/(x^2+1) = 0 
```
Subtract (3x(y-1))/(x^2+1) from both sides:
```
dy/dx = - (3x(y-1))/(x^2+1)
```
Separate the y terms and x terms:
```
dy/(y-1) = -3x/(x^2+1) dx
```

2. Integrating Both Sides

Now, we integrate both sides of the equation with respect to their respective variables:

Integrate the left side with respect to y:
```
∫ dy/(y-1) = ln|y-1| + C1 
```

Integrate the right side with respect to x:

∫ -3x/(x^2+1) dx = -3/2 * ln(x^2+1) + C2

3. Combining Constants and Finding the General Solution

Now, we have:

ln|y-1| = -3/2 * ln(x^2+1) + C

where C = C2 - C1.

To simplify, we can:

Exponentiate both sides:
```
|y-1| = e^(-3/2 * ln(x^2+1) + C)
```
Simplify using properties of exponents:
```
|y-1| = e^C * (x^2+1)^(-3/2) 
```
Introduce a new constant, K = e^C:
```
|y-1| = K * (x^2+1)^(-3/2) 
```
Remove absolute value by considering both positive and negative cases:
```
y-1 = ± K * (x^2+1)^(-3/2) 
```
Solve for y:
```
y = 1 ± K * (x^2+1)^(-3/2)
```

This is the general solution to the differential equation.

4. Conclusion

We have successfully solved the differential equation (x^2+1)dy/dx + 3x(y-1) = 0 using the method of separation of variables. The general solution is given by:

y = 1 ± K * (x^2+1)^(-3/2)

where K is an arbitrary constant.