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Results tagged with ordinary-differential-equations
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user 359649
For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.
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Approach in solving this First Order Nonlinear ODE
Given differential equation can be written as $$ \frac{dy}{dx}+\frac{y}{x}=\frac{1}{3xy^2} $$ which is Bernoulli Equation and after the substitution $y^3 = z$ can be converted into first order linear …
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0
answers
36
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Solution of Bernoulli equation with inhomogeneous term
Consider the differential equation
$$ \frac{dy}{dx}+ P(x)y = Q(x)y^{n} + R(x) .$$
If $R(x)=0$, then I know how to solve it but for a non zero sufficiently smooth function $R(x)$, is it possible to fin …
- 1,271
1
vote
Solution of the differential equation $(x^2\sin^3y – y^2 \cos x)dx + (x^3\cos y\sin^2y–2y\si...
$\textbf{Hint:} ~$
Observe that,
$$ x^{2} {\sin}^{3}{y} \text{dx}+ x^{3} \cos{y} ~{\sin}^{2}{y} \text{dy}=\frac{1}{3}d(x^{3} {\sin}^{3}{y}) $$
$$ y^{2} \cos{x} \text{dx} +2y \sin{x} \text{dy}=d(y^{2} …
- 1,271
1
vote
Accepted
Existence and Uniqueness Theorem in a situation where the y-derivative is unbounded
$\frac{f(x,y)-f(x,1)}{y-1}=x^{2}(y-1)^{\frac{-2}{3}}$, $y \ne 1$ which is unbounded in any rectangle $|x| \le a$, $|y-1| \le b$, where a and b are finite non negative real numbers. So, Lipschitz condi …
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2
votes
1
answer
1k
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Solution of a second order non linear ordinary differential equation
Consider the ordinary differential equation
$$ \frac{d^{2}y}{dx^{2}} = \alpha \sinh{y},~~ y = \beta~~\text{along}~x=-\frac{1}{2},x=\frac{1}{2}~~\text{and}~~\frac{dy}{dx}=0~~\text{along}~x=0.$$
Here $\ …
- 1,271
2
votes
Inhomogeneous Cauchy-Euler equation: $x^2y''(x)+xy'(x)-y(x)=x$ for $x>0$
I don't know the logic of Wolfram alpha, but both the solutions are same if you take constants $c_{1}=c_{1}-\frac{1}{4}+\frac{ic_{2}}{2}$ and $c_{2}=c_{1}-\frac{ic_{2}}{2}$
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3
votes
How to solve the following differential equation $(y^3-2yx^2)dx+(2xy^2-x^3)dy=0.$
Just observe that your differential equation is homogeneous, so substitution $y=vx$ will simplify it.
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3
votes
How to solve system of differential equations without using the time parameter $t$?
Differential equation $\frac{dy}{dx}=\frac{-y+x^2}{x} $ can be written as $$ \frac{dy}{dx} + \frac{y}{x}=x $$ which is standard linear non homogeneous differential equation of first order and be solve …
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3
votes
What is the reason behind trying to find the solution of a differential equation?
A differential equation involves unknown functions and derivatives. You are asking why we solve differential equations, I will give you some reasons: Suppose you want to find the area of some curve $$ …
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1
answer
67
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Solution of first order differential equation with integral type initial condition
Suppose we want to solve the ODE
$$ \frac{dx}{dt} = f(t)x(t)$$ with initial condition
$$x(0) = c_{1}+c_{2} \int_{0}^{\infty} g(t)x(t)dt $$
where $f$ and $g$ are continuous function.
Then is it possibl …
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1
vote
1
answer
97
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Constant solutions of $\cfrac{dy}{dx} = \cfrac{x^{3}-x}{1+e^{x}}$
Consider the differential equation
$\cfrac{dy}{dx} = \cfrac{x^{3}-x}{1+e^{x}}$. How to find constant solution $y(x)$ and $\displaystyle\lim_{x \to \infty}y(x),$ where $y(x)$ is the solution such that …
- 1,271
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1
answer
134
views
How to find integral curves to a given system of ODEs?
How to find integral curves to a given system of ODEs
$$ \frac{dx}{y+3z}=\frac{dy}{z+5x} = \frac{dz}{x+7y}. $$
I tried to find multipliers but did not succeed. Can someone help to find integral cur …
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votes
1
answer
86
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Solution of first order integro-differential equation
Consider
$$ \frac{dy}{dx} + p(x)y = \int_{0}^{\infty} y(x)dx = R~~(say) $$
$$ y(0) = \int_{0}^{\infty} f(x)y(x)dx.$$
I want to solve above differential equation.
Here is my try:
$$ y(x) = y(0) e^{-\in …
- 1,271
1
vote
1
answer
205
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Solving first order differential equation with integral term
Consider
$$ \frac{dy}{dx} + p(x)y = \int_{0}^{\infty} y(x)dx. $$
I want to solve above differential equation. Can I consider right hand side as constant to solve this?
I know RHS is a constant but it …
- 1,271
0
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Should the derivative of a function correspond to its differential equation?
Clearly, solution $y=\frac{2}{3}x + \frac{17}{9} $ is satisfying your ODE. May be you are forgetting to substutute y on right hand side.
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