📘 differential equations

1. مسئله: معادله دیفرانسیل $$ (x+1)y'' - (2x+3)y' + (x+2)y = 0 $$ داده شده است و یک جواب خاص آن $$ y_1 = e^x $$ است. هدف یافتن جواب عمومی معادله است. 2. فرمول و روش: چون یک جواب خا

1. مسئله: معادله دیفرانسیل $$2x^2 y'' + 3xy' - (1+x)y = 0$$ را به صورت سری توانی حول نقطه $$x_0=0$$ حل کنیم. 2. فرض می‌کنیم جواب به صورت سری توانی باشد:

1. **State the problem:** Solve the differential equation $$2 - x''xe^y + (4x - 2x^2)e^x y' + e^{-x} (2 - 4x + x^2)y = e^x - x.$$ 2. **Rewrite the equation clearly:** The equation

1. **State the problem:** We need to solve the integro-differential equation

1. مسئله: حل معادله دیفرانسیل $$x^2 e^{-\pi} y'' + (4x - 2x^2) e^{-x} y' + e^{-\pi} (2 - 4x + x^2) y = e^{x} - x.$$ 2. ابتدا معادله را ساده می‌کنیم. توجه کنید که $$e^{-\pi}$$ ضریب

1. **State the problem:** We are given the differential equation $$y'' - y = 0$$ with the general solution $$y = c_1 e^x + c_2 e^{-x}$$.

1. **State the problem:** We want to approximate the solution to the initial value problem $$y' = \frac{3}{x}(y^2 + y), \quad y(6) = 3$$ at points $x = 6.2, 6.4, 6.6, 6.8$ using Eu

1. **Problem statement:** Verify that the straight lines $y = \pm \frac{5}{3}x$ are solution curves to the differential equation $$\frac{dy}{dx} = \frac{25x}{9y}$$ for $x \neq 0$.

1. **State the problem:** Rewrite the implicit solution $$-\frac{y}{x} - \ln\left|1 - \frac{y}{x}\right| = \ln|x| + C$$

1. **State the problem:** Solve the differential equation $$\frac{dv}{dx} = \frac{x^2 - xy + y^2}{xy}$$ by substituting $$y = vx$$. 2. **Substitution:** Given $$y = vx$$, then $$\f

1. **State the problem:** Solve the integral differential equation $$y''(x) - e^x \int_0^x e^{-t} y''(t) \, dt = y(x) + u_1(x)$$

1. **State the problem:** Solve the differential equation $$xy'' - (2 + x)y' + 4y = 0$$ with initial condition $$y(0) = 0$$ using the Laplace transform. 2. **Rewrite the equation:*

1. **State the problem:** Solve the differential equation $$xy'' - (2 + x)y' + 4y = 0$$ with the initial condition $$y(0) = 0$$. 2. **Identify the type of equation:** This is a lin

1. **State the problem:** Solve the differential equation $$xy'' - (2 + x)y' + 4y = 0$$ with initial condition $$y(0) = 0$$ using the Laplace transform. 2. **Rewrite the equation:*

1. **State the problem:** We need to solve the integral equation $$y'(x) = x^2 + \int_0^x y(t) \cos(x - t) \, dt$$ with the initial condition $$y(0) = 4$$. 2. **Recognize the type

1. مسئله را بیان می‌کنیم: معادله دیفرانسیل انتگرالی داده شده است: $$y'(x) = x^2 + \int_0^x y(t) \cos(x - t) \, dt, \quad y(0) = 4$$

1. مسئله را بیان می‌کنیم: معادله دیفرانسیل انتگرالی داده شده است: $$y'(x) = x^2 + \int_0^x y(t) \cos(x - t) \, dt, \quad y(0) = 4$$

1. **State the problem:** Solve the differential equation $$y'' + 2y' + y = xe^{-x}$$ with initial conditions $$y(0) = 1$$ and $$y'(0) = -2$$ using the Laplace transform. 2. **Appl

1. **State the problem:** A radioactive substance decays at a rate proportional to its mass. When the mass is 26 mg, the decay rate is 10 mg per week. We want to find a formula for

1. Statement of the problem. Problem: Solve the ordinary differential equation using the Laplace transform method.

1. **State the problem:** Solve the second order ODE $$\frac{d^2y}{dt^2} - 6\frac{dy}{dt} + 16y = 2e^{2t}$$ with initial conditions $$y(0) = -1$$ and $$y'(0) = -4$$ using Laplace t