Problem 33 - Advanced Engineering Mathematics

Question

We note that \(\left(N_{x}-M_{y}\right) / M=2 / y,\) so an integrating factor is \(e^{\int 2 d y / y}=y^{2}\). Let \(M=6 x y^{3}\) and \(N=4 y^{3}+9 x^{2} y^{2}\) so that \(M_{y}=18 x y^{2}=N_{x} .\) From \(f_{x}=6 x y^{3}\) we obtain \(f=3 x^{2} y^{3}+h(y), h^{\prime}(y)=4 y^{3},\) and \(h(y)=y^{4} .\) A solution of the differential equation is \(3 x^{2} y^{3}+y^{4}=c\)

Step-by-Step Solution

Answer

The solution of the differential equation is \(3x^2y^3 + y^4 = c\).

Step 1: Identify Given Information

We are given a differential equation in the form of \((N_x - M_y) / M = 2 / y\). We know the functions \(M\) and \(N\): \(M = 6xy^3\) and \(N = 4y^3 + 9x^2y^2\). We are also informed that \(M_y = 18xy^2 = N_x\).

Step 2: Determine the Integrating Factor

We calculate the integrating factor, which is given by \(e^{\int \frac{2}{y} \cdot dy}\). Simplifying the integral, we have \(\int \frac{2}{y} \, dy = 2 \ln y = \ln (y^2)\). Thus, the integrating factor is \(e^{\ln(y^2)} = y^2\).

Step 3: Verify Condition for Exactness

The differential equation is exact if \(M_y = N_x\). In our case, \(M_y = 18xy^2\) and \(N_x = 18xy^2\), which confirms the equation is exact.

Step 4: Integrate with Respect to x

From \(f_x = 6xy^3\), we integrate with respect to \(x\) to find \(f(x, y)\). The integral of \(6xy^3\) with respect to \(x\) is \(3x^2y^3 + h(y)\), where \(h(y)\) is an unknown function of \(y\).

Step 5: Determine h(y)

Differentiate \(f(x, y)\) with respect to \(y\) to find \(h'(y)\). Knowing that \(N = 4y^3 + 9x^2y^2\), we set the equation \(3x^2y^3 + h'(y) = 4y^3 + 9x^2y^2\). Comparing terms that only depend on \(y\), we find that \(h'(y) = 4y^3\).

Step 6: Integrate h'(y)

To find \(h(y)\), integrate \(4y^3\) with respect to \(y\), which gives \(h(y) = y^4 + C\), where \(C\) is an arbitrary constant.

Step 7: Formulate the Solution

Now we can write the general solution of the differential equation. Combining results, the solution is \(3x^2y^3 + y^4 = C\), where \(C\) is a constant.

Key Concepts

Integrating Factor

In solving differential equations, an integrating factor is a function that, when multiplied by a given differential equation, makes it easier to solve. Specifically, it transforms a non-exact equation into an exact one.
An integrating factor is particularly useful if you have an equation of the form \[ \frac{N_x - M_y}{M} = rac{2}{y} \]In this case, we identify the integrating factor by evaluating \[ e^{\int \frac{2}{y}\, dy} \]Simplifying the integral \( \int \frac{2}{y} \, dy = 2 \ln y = \ln(y^2) \),leads to an integrating factor of \( y^2 \).

• This step is crucial to make the differential equation solvable.
• The same procedure can be applied generally depending on the form of the original equation.

Exactness Condition

An exact differential equation is one where \( M_y = N_x \).For our specific problem, the functions \( M = 6xy^3 \)and \( N = 4y^3 + 9x^2y^2 \),satisfy this condition because \( M_y = 18xy^2 \)is equal to \( N_x = 18xy^2 \).

• This equality is checked before proceeding. It confirms that the integrating factor effectively converts the equation to a solvable form.
• Exactness assures us that there exists a potential function whose differentials lead back to the original equation.

Understanding the exactness condition helps us determine the simplest mathematical path to solve the differential equation.

Differential Equation Solution

Solving a differential equation involves integrating to find a potential function. For this exercise, we start with acknowledging \( f_x = 6xy^3 \) and must integrate with respect to \( x \):\[ f(x, y) = \int 6xy^3 \, dx = 3x^2y^3 + h(y) \]where \( h(y) \) is a function that depends solely on \( y \).

• This step extracts the structure of the solution by separating variables and integrating each part.
• The presence of \( h(y) \) allows us flexibility to incorporate partial derivatives attributed to \( y \).

This approach effectively combines known solutions and any additional variable-dependent terms into a coherent solution.

Mathematical Integration

Mathematical integration plays a pivotal role in forming the final solution of a differential equation. Once \( f(x, y) = 3x^2y^3 + h(y) \) was derived,

Determine \( h(y) \)

We needed to differentiate \( f(x, y) \) with respect to \( y \) to find \( h'(y) \).With \( N = 4y^3 + 9x^2y^2 \):Comparing, we derive \( h'(y) \) = 4y^3,integrating this,\[ h(y) = \int 4y^3 \, dy = y^4 + C \]

• Integration determines the unknown part \( h(y) \) and completes the potential function.
• It provides consistency by ensuring that all parts contribute to a balanced equation.
• The constant \( C \) represents any constant of integration that can arise in indefinite integrals.

Finally, by putting all parts together, we arrived at the complete solution: \( 3x^2y^3 + y^4 = C \).