Free preview of The Ultimate Crash Course for STEM Majors exact differential equations test response

Exact Differential Equations Test Response: Verifying the Exact Theorem and Final Solution

This lesson is nothing more than a free preview of The Ultimate Crash Course for STEM Majors. This worked differential equations example shows how to organize a test response, apply the Exact Theorem, integrate the potential function, state the final implicit solution, and prepare to verify that the answer satisfies the original ordinary differential equation.

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NOTE A constant with a constant is a constant. \(-c\) is a variation of \(C\). \(\psi=0\) in this case. Depending on the course you are in, you will choose to label. We are master of labeling.

TEST RESPONSE

QUESTION Solve the ODE

\[ \left(\frac{y}{x}+6x\right)+(\color{red}{\ln x}-2)y’=0, \qquad x>0. \]

(Step 1) By exact equation \(M+Ny’=0\), if \(\color{#00a651}{M_y=N_x}\), then it is exact.

(Step 2)

\[ \psi_x=M(x,y)=\frac{y}{x}+6x, \qquad \psi_y=N(x,y)=\ln x-2. \]

\[ \Rightarrow\quad \psi=\int \frac{y}{x}+6x\,dx, \qquad \psi=\int \ln x-2\,dy \]

\[ \Rightarrow\quad \psi=\int \frac{y}{x}\,dx+\int 6x\,dx, \qquad \psi=\int \ln x\,dy-\int 2\,dy \]

\[ \Rightarrow\quad \psi=y\int\frac{1}{x}\,dx+6\int x\,dx, \qquad \psi=\ln x\int dy-2\int dy \]

\[ \Rightarrow\quad \psi=y\ln|x|+6\left[\frac{1}{2}x^2\right]+g_1(y), \qquad \psi=\ln x[y]-2[y]+g_2(x) \]

\[ \Rightarrow\quad \psi=y\ln|x|+3x^2+\color{red}{g_1(y)}, \qquad \psi=y\ln x-2y+g_2(x) \]

\[ \Rightarrow\quad \psi=y\ln|x|+3x^2+\color{red}{g_1(y)}, \qquad \psi=y\ln x+\color{red}{(-2y)}+g_2(x) \]

\[ \Rightarrow\quad \psi=y\ln|x|+\color{#00a651}{3x^2}+\color{red}{g_1(y)}, \qquad \psi=y\ln x+\color{red}{(-2y)}+\color{#00a651}{g_2(x)}. \]

\(\mathcal{F}_3\) Finalize and state the answer with citation.

By “The Exact Theorem”

\[ y\ln x+3x^2-2y=c. \]

[FC] How do we know this is right? The book matches but how do we know?

It is now your turn to show the answer satisfies the question.

[RP] Find \(f(x)\), provided the vector function, \[ \mathbf{F}= \left\langle \frac{y}{x}+6x,\ln x-2 \right\rangle. \] Use techniques either from book [7] or [36]. Make sure to stick with their notations, definitions, and theorems. Type it up and add it to the comment box via MathMonthly.com .

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