Math / Calculus

Question16. $e^{y}-x^{2} y=0$ .

Studdy Solution

STEP 1

Assumptions
1. The given equation is $e^{y} - x^{2} y = 0$ .
2. We are solving for $y$ in terms of $x$ .
3. The equation involves the exponential function $e^y$ and a polynomial term $x^2 y$ .
4. We assume $y$ is a differentiable function of $x$ if needed for further analysis.

STEP 2

Rearrange the equation to isolate the exponential term on one side.
$e^{y} = x^{2} y$

STEP 3

To solve for $y$ , we can attempt to express $y$ explicitly in terms of $x$ . However, this equation is transcendental, meaning it involves both algebraic and transcendental functions, which often do not have a simple closed-form solution.

STEP 4

Consider the possibility of using numerical methods or graphical analysis to find $y$ for specific values of $x$ . For analytical insight, let's explore the behavior of the equation.

STEP 5

If we consider $y$ as a function of $x$ , denoted as $y(x)$ , we can analyze the equation by considering special cases or approximations. For instance, if $x = 0$ , then:
$e^{y} = 0 \cdot y = 0$
This is not possible since $e^y$ is never zero for real $y$ . Therefore, $x = 0$ does not provide a meaningful solution.

STEP 6

For small values of $x$ , we can use a series expansion or approximation methods. However, the equation $e^{y} = x^{2} y$ suggests that $y$ must be positive for $x \neq 0$ to balance the equation.

STEP 7

Consider the case where $y$ is small, and use the approximation $e^y \approx 1 + y$ for small $y$ . Substitute this into the equation:
$1 + y \approx x^2 y$

STEP 8

Rearrange the approximate equation to solve for $y$ :
$1 \approx x^2 y - y$
$1 \approx y(x^2 - 1)$

STEP 9

Solve for $y$ in terms of $x$ :
$y \approx \frac{1}{x^2 - 1}$
This approximation is valid for small $y$ and when $x^2 \neq 1$ .

STEP 10

The solution $y \approx \frac{1}{x^2 - 1}$ is an approximation and provides insight into the behavior of $y$ for small values. For exact solutions, numerical methods or specific tools may be necessary to solve the transcendental equation for given $x$ .
Since the problem does not specify a particular method or further conditions, this is the extent of the analytical solution.

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Question16. $e^{y}-x^{2} y=0$ .

Studdy Solution

STEP 1

Assumptions
1. The given equation is $e^{y} - x^{2} y = 0$ .
2. We are solving for $y$ in terms of $x$ .
3. The equation involves the exponential function $e^y$ and a polynomial term $x^2 y$ .
4. We assume $y$ is a differentiable function of $x$ if needed for further analysis.

STEP 2

Rearrange the equation to isolate the exponential term on one side.
$e^{y} = x^{2} y$

STEP 3

To solve for $y$ , we can attempt to express $y$ explicitly in terms of $x$ . However, this equation is transcendental, meaning it involves both algebraic and transcendental functions, which often do not have a simple closed-form solution.

STEP 4

Consider the possibility of using numerical methods or graphical analysis to find $y$ for specific values of $x$ . For analytical insight, let's explore the behavior of the equation.

STEP 5

STEP 6

For small values of $x$ , we can use a series expansion or approximation methods. However, the equation $e^{y} = x^{2} y$ suggests that $y$ must be positive for $x \neq 0$ to balance the equation.

STEP 7

Consider the case where $y$ is small, and use the approximation $e^y \approx 1 + y$ for small $y$ . Substitute this into the equation:
$1 + y \approx x^2 y$

STEP 8

Rearrange the approximate equation to solve for $y$ :
$1 \approx x^2 y - y$
$1 \approx y(x^2 - 1)$

STEP 9

Solve for $y$ in terms of $x$ :
$y \approx \frac{1}{x^2 - 1}$
This approximation is valid for small $y$ and when $x^2 \neq 1$ .

STEP 10

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Question16. ey−x2y=0e^{y}-x^{2} y=0ey−x2y=0.

Studdy Solution

STEP 1

Assumptions1. The given equation is ey−x2y=0e^{y} - x^{2} y = 0ey−x2y=0.2. We are solving for yyy in terms of xxx.3. The equation involves the exponential function eye^yey and a polynomial term x2yx^2 yx2y.4. We assume yyy is a differentiable function of xxx if needed for further analysis.

STEP 2

Rearrange the equation to isolate the exponential term on one side. ey=x2ye^{y} = x^{2} yey=x2y

STEP 3

To solve for yyy, we can attempt to express yyy explicitly in terms of xxx. However, this equation is transcendental, meaning it involves both algebraic and transcendental functions, which often do not have a simple closed-form solution.

STEP 4

Consider the possibility of using numerical methods or graphical analysis to find yyy for specific values of xxx. For analytical insight, let's explore the behavior of the equation.

STEP 5

STEP 6

For small values of xxx, we can use a series expansion or approximation methods. However, the equation ey=x2ye^{y} = x^{2} yey=x2y suggests that yyy must be positive for x≠0x \neq 0x=0 to balance the equation.

STEP 7

Consider the case where yyy is small, and use the approximation ey≈1+ye^y \approx 1 + yey≈1+y for small yyy. Substitute this into the equation:1+y≈x2y1 + y \approx x^2 y1+y≈x2y

STEP 8

Rearrange the approximate equation to solve for yyy:1≈x2y−y1 \approx x^2 y - y1≈x2y−y1≈y(x2−1)1 \approx y(x^2 - 1)1≈y(x2−1)

STEP 9

Solve for yyy in terms of xxx:y≈1x2−1y \approx \frac{1}{x^2 - 1}y≈x2−11​This approximation is valid for small yyy and when x2≠1x^2 \neq 1x2=1.

STEP 10

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Question16. $e^{y}-x^{2} y=0$ .

Assumptions
1. The given equation is $e^{y} - x^{2} y = 0$ .
2. We are solving for $y$ in terms of $x$ .
3. The equation involves the exponential function $e^y$ and a polynomial term $x^2 y$ .
4. We assume $y$ is a differentiable function of $x$ if needed for further analysis.

Rearrange the equation to isolate the exponential term on one side.
$e^{y} = x^{2} y$

To solve for $y$ , we can attempt to express $y$ explicitly in terms of $x$ . However, this equation is transcendental, meaning it involves both algebraic and transcendental functions, which often do not have a simple closed-form solution.

Consider the possibility of using numerical methods or graphical analysis to find $y$ for specific values of $x$ . For analytical insight, let's explore the behavior of the equation.

For small values of $x$ , we can use a series expansion or approximation methods. However, the equation $e^{y} = x^{2} y$ suggests that $y$ must be positive for $x \neq 0$ to balance the equation.

Consider the case where $y$ is small, and use the approximation $e^y \approx 1 + y$ for small $y$ . Substitute this into the equation:
$1 + y \approx x^2 y$

Rearrange the approximate equation to solve for $y$ :
$1 \approx x^2 y - y$
$1 \approx y(x^2 - 1)$

Solve for $y$ in terms of $x$ :
$y \approx \frac{1}{x^2 - 1}$
This approximation is valid for small $y$ and when $x^2 \neq 1$ .