Solving the Differential Equation: (x^(3)+x^(2)+x+1)(dy)/(dx)=2x^(2)+x
Given the differential equation:
$(x^{3}+x^{2}+x+1)\frac{dy}{dx}=2x^{2}+x$
with the initial condition:
$y=1 \quad when \quad x=0$
Step 1: Simplify the Differential Equation
To start, we can simplify the left-hand side of the equation by expanding the product:
$x^{3}\frac{dy}{dx}+x^{2}\frac{dy}{dx}+x\frac{dy}{dx}+\frac{dy}{dx}=2x^{2}+x$
Step 2: Rearrange the Terms
Next, we can rearrange the terms to get:
$\frac{dy}{dx}(x^{3}+x^{2}+x+1)=2x^{2}+x$
Step 3: Divide Both Sides
Now, we can divide both sides of the equation by $(x^{3}+x^{2}+x+1)$ to get:
$\frac{dy}{dx}=\frac{2x^{2}+x}{x^{3}+x^{2}+x+1}$
Step 4: Integrate Both Sides
To solve for $y$, we can integrate both sides of the equation with respect to $x$:
$\int\frac{dy}{dx} dx=\int\frac{2x^{2}+x}{x^{3}+x^{2}+x+1} dx$
Step 5: Evaluate the Integral
Evaluating the integral on the right-hand side, we get:
$y=\ln|x^{3}+x^{2}+x+1|+C$
where $C$ is the constant of integration.
Step 6: Apply the Initial Condition
Using the initial condition $y=1$ when $x=0$, we can find the value of $C$:
$1=\ln|0^{3}+0^{2}+0+1|+C$
$C=1$
Step 7: Write the Final Solution
Finally, we can write the final solution as:
$y=\ln|x^{3}+x^{2}+x+1|+1$
Thus, we have solved the differential equation and found the general solution.