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Chapter 41: Problem 1079

Find the general solution of: \((d y / d x)=\left[\left(x^{2}+y^{2}+y\right) /(x)\right]\).

Short Answer

Expert verified
The general solution for the given first-order differential equation is: \(y(x) = \frac{x^2y}{2} - \frac{y^3}{3} - \frac{y^2}{2} + C\), which is an implicit solution.

Step by step solution

01

Rewrite the differential equation in a separable form

We are given the differential equation: \(\frac{dy}{dx} = \frac{x^2 + y^2 + y}{x}\) We need to manipulate the equation to separate the variables, x and y. Multiply both sides by x to get: \(x \frac{dy}{dx} = x^2 + y^2 + y\) Now, move all terms with x to one side and all terms with y to the other side: \(x dy - (y^2 + y)dx = 0\) Now, we have a separable form: \(x dy = (y^2 + y)dx\)
02

Integrate both sides of the separable equation

Now we need to integrate both sides of the equation. Integrate the left side with respect to y: \[\int x dy = \int ( y^2 + y) dx\] The left side of the equation is easy to integrate, giving: \(\frac{1}{2}x^2y = \int (y^2 + y)dx \) Now, integrate the right side of the equation with respect to x. Remember to leave the constant of integration: \(\frac{1}{2}x^2y = \frac{1}{3}y^3 + \frac{1}{2}y^2 + C\)
03

Solve for the general solution

Now, we rearrange the equation to make it a general solution for the given differential equation: \(y(x) = \frac{x^2y}{2} - \frac{y^3}{3} - \frac{y^2}{2} + C\) This is the general solution for the given first-order differential equation. Note that this general solution is implicit, meaning that the equation isn't explicitly solved for y in terms of x. However, in many cases, particularly for higher-order differential equations, finding an implicit solution is the best we can do.

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Most popular questions from this chapter

Solve the differential equations: \(\left(\mathrm{y}^{\prime}+\mathrm{y}\right)=\mathrm{x}\).

Solve the differential equation: \(\left[\left(\mathrm{d}^{2} \mathrm{y}\right) /\left(\mathrm{dx}^{2}\right)\right]=\cos 2 \mathrm{x}\)

Find the general solution for: \((\mathrm{dy} / \mathrm{dx})=[(\mathrm{x}-2 \mathrm{y}) /(2 \mathrm{x}-\mathrm{y})]\).

Find the general solution of \(\left(\mathrm{D}^{2}-1\right) \mathrm{y}=2 \mathrm{x}+\mathrm{e}^{2 \mathrm{x}}\).

Solve the differential equation: \(\mathrm{y}^{\prime \prime}-2 \mathrm{y}^{\prime}-\mathrm{y}=0\).
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