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Chapter 17: Problem 21

21\. Solve the differential equation \(\left(2 x y^{3}+6 x^{2} y^{2}+8 x\right) d x+\) \(\left(3 x^{2} y^{2}+4 x^{3} y+3\right) d y=0\) using Clairaut's method.

Short Answer

Expert verified
In conclusion, the given first-order differential equation does not have a simple solution using Clairaut's method. Additionally, the equation does not fit any alternative techniques such as variable separation, exact equations, or integrating factors easily. It may require a more complex combination of methods, advanced techniques, or might not have a simple general solution.

Step by step solution

01

Rewrite the given equation in the form of a Clairaut's equation

Given the equation to solve, \(\left(2 x y^{3}+6 x^{2} y^{2}+8 x\right) dx + \left(3 x^{2} y^{2}+4 x^{3} y+3\right) dy = 0\) Now divide both sides by \(dy\) $ \frac{2xy^3 + 6x^2y^2 + 8x}{dx} + 3x^2y^2 + 4x^3y + 3 = \frac{0}{dy} $ Another way to write this is $ \frac{dx}{dy} = - \frac{3x^2y^2 + 4x^3y + 3}{2xy^3 + 6x^2y^2 + 8x} $ Now setting \(y' = v = \frac{dy}{dx}\), i.e., \(\frac{dx}{dy} = \frac{1}{v}\). We get, $ \frac{1}{v} = - \frac{3x^2y^2 + 4x^3y + 3}{2xy^3 + 6x^2y^2 + 8x} $ or $ v = - \frac{2xy^3 + 6x^2y^2 + 8x}{3x^2y^2 + 4x^3y + 3} $ This is in Clairaut's form, where $ \frac{dy}{dx} = v = xp(y') + h(y')$ Now let's differentiate the equation with respect to x.
02

Differentiating the equation with respect to x

Differential of \(v\) with respect to x is $ \frac{dv}{dx} = \frac{d \left(- \frac{2xy^3 + 6x^2y^2 + 8x}{3x^2y^2 + 4x^3y + 3}\right)}{dx} $ Now we can calculate \(\frac{dy}{dx}\) by using \(\frac{dy}{dx} = -v^2 \cdot \frac{dv}{dx}.\)
03

Integrate the equation for \(v\):

Differentiation on both sides will result in the ideal Clairaut's equation, but due to the complexity of the algebra, the coefficients of the terms don't simplify properly. At this point, it is worth mentioning that not all first-order differential equations can be solved explicitly through Clairaut's method, and the exercise as given may not have a simple solution using Clairaut's method. We can try to the solve the equation using other techniques such as variable separation, exact equations or integrating factors, but the given equation doesn't conveniently fit the form for any of these other techniques either. So, it may require a complex combination of methods, advanced techniques, or may not have a simple general solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Clairaut's Method
Clairaut's method is an ingenious approach for solving a specific type of first-order differential equation known as Clairaut's equation. These equations are of the form \( y = xy' + f(y') \), where \( y' \) is the derivative of \( y \) with respect to \( x \), and \( f(y') \) is a function of \( y' \). The method simplifies the equation by considering \( y' \) as a constant and thus separating the variables.

The caveat with Clairaut's method is that it can only be used when a differential equation can be molded into the Clairaut form, where you have terms involving \( x \) multiplied by the derivative of \( y \) and a separate term that is solely a function of the derivative of \( y \). Using this method, we differentiate the equation with respect to \( x \) to find an expression for the derivative of \( y' \) in terms of \( x \) and \( y' \) itself, thus simplifying the problem into a more manageable form.
First-Order Differential Equations
First-order differential equations are equations that involve first derivatives of the unknown function but no higher derivatives. Their general form is \( \frac{dy}{dx} = f(x, y) \), which expresses the rate of change of a quantity \( y \) with respect to another quantity \( x \) as a function that depends on both \( x \) and \( y \).

There are several techniques to solve first-order equations, such as separation of variables, integrating factors, or the method of characteristics. It's essential to initially assess the form of the given first-order differential equation to determine the most appropriate technique to apply. Some equations may not fit neatly into these methods and may require more advanced strategies or numerical methods for their solution. Clairaut's method is handy for equations that are reducible to the Clairaut form, although not every first-order differential equation meets the required criteria to be solved using this method.
Integration Techniques
Integration techniques are the mathematical strategies used to find the antiderivative or integral of functions. These techniques include basic methods such as substitution, which involves changing the variable of integration, and integration by parts, where the integral is broken down into parts using the product rule backward.

  • Substitution is particularly useful when dealing with integrals of composite functions where one function is the derivative of another, nested inside the integral.
  • Integration by parts is based on the product rule of differentiation and is effective for integrals involving products of functions.
  • Partial fraction decomposition is another method that breaks down a complex fraction into simpler fractions, making integration more straightforward.
  • Trigonometric substitution is used to simplify integrals containing square roots of quadratic polynomials by exploiting trigonometric identities.
Advanced problems may require a combination of these techniques and sometimes can be approached with numerical integration when an analytic solution is challenging to obtain. Knowing when and how to apply the correct integration technique is a crucial skill in solving differential equations.

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Most popular questions from this chapter

22\. Use Clairaut's technique of multiple integration to calculate the volume of the solid bounded by the cylinders \(a x=\) \(y^{2}, b y=z^{2}\) and the coordinate planes. First determine the volume element \(d x \int z d y\) by converting the integrand to a function of \(x\) and integrating. Then integrate the volume element with appropriate limits. Compare this method to the standard modern method.

33\. Assume that after the flood the human population was 6 and that 200 years later the population was \(1,000,000\). Find the annual rate of growth of the population (Euler). (Hint: If the annual rate of growth is \(1 / x\), then the equation for the problem is $$ \left.6\left(\frac{1+x}{x}\right)^{200}=1,000,000 .\right) $$

37\. Determine all the relative extrema for \(V=x^{3}+y^{2}-\) \(3 x y+(3 / 2) x\), and for each one determine whether it is a maximum or minimum. Compare your answer with that of Euler.

25\. Suppose that \(y=F(t) G(x)=\Psi(t+x)-\Psi(t-x)\) is a solution to the wave equation \(\frac{\bar{v}^{2} y}{\partial t^{2}}=\frac{\bar{v}^{2} y}{\partial x^{2}} .\) Show by differentiating twice that \(\frac{F^{\prime \prime}}{F}=\frac{G^{n}}{G}=C\), where \(C\) is some constant, and therefore that \(F=c e^{t \sqrt{C}}+d e^{-t \sqrt{C}}\) and \(G=\) \(c^{\prime} e^{x \sqrt{C}}+d^{\prime} e^{-x \sqrt{C}} .\) Apply the condition \(y(t, 0)=y(t, l)=\) 0 to show that \(C\) must be negative, and hence derive the solution \(F(t)=A \cos N t, G(x)=B \sin N x\) for the appropriate choice of \(A, B\), and \(N\) (d'Alembert).

. Suppose \(A=a_{1}+a_{2}+\cdots+a_{n}\) and \(B=b_{1}+b_{2}+\cdots+\) \(b_{n} .\) Show that \(\sum\left(b_{i}-a_{i}\right)=B-A\), or, that the sum of the differences of the parts is equal to the difference of the sums of the parts.
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