prompt
stringlengths 115
1.09k
| solution
stringlengths 13
300
| parameters
stringclasses 24
values | type
stringclasses 7
values | index
int64 0
82
|
|---|---|---|---|---|
Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = \frac{e^{2x}}{(1+e^x)^4} y$ with initial conditions at $y(0)=0, y'(0)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.
|
$\boxed{y(x) \sim 4\epsilon \cosh(x/2)\sinh\left[\frac{e^x-1}{2\epsilon(1+e^x)}\right]}$
|
$x; \epsilon$
|
wkb
| 18
|
Find the leading order WKB approximation for the specific differential equation: $\epsilon^2y''=x^2y$ with initial conditions at $y(1)=0; y(2)=1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.
|
$\boxed{y=\frac{\sqrt{2}}{\sqrt{x}}(exp(-\frac{2}{\epsilon})exp(\frac{x^2}{2\epsilon})-exp(-\frac{1}{\epsilon})exp(-\frac{x^2}{2\epsilon}))}$
|
$x; \epsilon$
|
wkb
| 19
|
Find the leading order WKB approximation for the specific differential equation: $\epsilon y'' + y = 0$ with initial conditions at $y(0) = 0, y(1) = 1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.
|
$\boxed{ \frac{\sin \left(\frac{x}{\sqrt{\epsilon}}\right)}{\sin \left(\frac{1}{\sqrt{\epsilon}}\right)}}$
|
$x; \epsilon$
|
wkb
| 20
|
Find the leading order WKB approximation for the specific differential equation: $\epsilon y''' + y = 0$ with initial conditions at $y(0) = 1, y(\epsilon^{1/3}) = e^{-1}, y(-\epsilon^{1/3}) = e^{1} $ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.
|
$\boxed{e^{\frac{-x}{\epsilon^{1/3}}}}$
|
$x; \epsilon$
|
wkb
| 21
|
Find the leading order behavior of $\epsilon^2 y'' + y = 0$ subject to boundary conditions $y(0)=0, y(1)=1$ in the limit of small $\epsilon$. Give your final answer in a \boxed latex environment.
|
$\boxed{sin(x/\epsilon)/sin(1/\epsilon)}$
|
$x; \epsilon$
|
wkb
| 22
|
Find the leading order approximation of $y(x)$ from the differential equation $\epsilon^2 y''(x)=(\sin x) y$ subject to boundary conditions $y(\frac{\pi}{2}) = 1, y'(\frac{\pi}{2}) = 0$ in the limit of $\epsilon \to 0$. The answer should be in terms of the incomplete beta function $B_z(a, b)$ where $B_z(a, b)=\int_0^z t^{a-1}(1-t)^{b-1} d t, 0 \leq z \leq 1$
|
$\boxed{\frac{1}{(\sin x)^{\frac{1}{4}}} \cosh \left(\frac{1}{2 \epsilon}\left[B_{(\sin(x))^2}\left(\frac{3}{4}, \frac{1}{2}\right)-B\left(\frac{3}{4}, \frac{1}{2}\right)\right]\right)}$.$
|
$x; \epsilon$
|
wkb
| 23
|
Find the leading order behavior of $y''=(\cot x)^4 y$ in the limit of small x that satisfies the conditions $y(1) = 1; y'(1) = 1$. Leading order does not necessarily mean only one term. Use only the variables and constants given in the problem; do not define additional constants. Give your final answer in a \boxed latex environment.
|
$\boxed{y(x) \sim \frac{e}{2} x e^{-1/x} + \frac{1}{2e} x e^{1/x}}$
|
$x$
|
wkb
| 24
|
Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' = y \cosh^2 x$ with initial conditions at $y(0) = 0; y'(0) = 1$ where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.
|
$\boxed{y(x) = \epsilon (\cosh x)^{-1/2} \sinh\left( \frac{1}{\epsilon} \sinh x \right)}$
|
$x; \epsilon$
|
wkb
| 25
|
Find the leading order WKB approximation for the specific differential equation: $\epsilon^2 y'' + \cosh(x) y' + \sinh(x) y = 0$ with initial conditions at $y(0) = 0$, $y(1) = 1$. where $\epsilon$ is a small positive parameter ($0 < \epsilon \ll 1)$. Give your final answer in a \boxed latex environment.
|
$\boxed{y(x) = \cosh(1)(\frac{1}{\cosh(x)} - \cosh(x) e^{-\sinh(x)/\epsilon^2}) }$
|
$x; \epsilon$
|
wkb
| 26
|
Find the WKB approximation up to the second leading order for the specific differential equation as $x\to 0$ $3x^5y'''=y$ with initial conditions at $y(1)=1, y'(1)=0, y''(1)=0$ as $x\to 0$. Give ONE final answer in terms of y and x and numbers (not arbitrary constants) in a $\boxed{}$ latex environment.
|
$\boxed{y(x) \approx x^{5/3} \left( 1.9461391296842614 e^{-\frac{3^{2/3}}{2}x^{-2/3}} + e^{\frac{3^{2/3}}{4}x^{-2/3}}\left(-1.4641592737823024 \cos\left(\frac{3^{7/6}}{4}x^{-2/3}\right) + 1.3969897071670665 \sin\left(\frac{3^{7/6}}{4}x^{-2/3}\right) \right) \right)}$
|
$x$
|
wkb
| 27
|
Find the WKB approximation up to the second leading order for the specific differential equation as $x\to 0$ $x^6 y''' + y = 0$ with initial conditions at $y(1)=1, y'(1)=0, y''(1)=0$ as $x\to 0$. Give ONE final answer in terms of y and x and numbers in a $\boxed{}$ latex environment.
|
$\boxed{y(x) \approx x^2 \left( 0.6131324019838598 e^{x^{-1}} + e^{-x^{-1}/2}\left(-0.712093006628941 \cos\left(\frac{\sqrt{3}}{2}x^{-1}\right) -0.8372865406850002 \sin\left(\frac{\sqrt{3}}{2}x^{-1}\right) \right) \right)}$
|
$x$
|
wkb
| 28
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.