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Differential Equation Solvers Calculator

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Example
Created on 2024-06-20Asked by James Smith (Solvelet student)
Solve the differential equation dydx=2x \frac{dy}{dx} = 2x with initial condition y(0)=3 y(0) = 3 .

Solution

Step 1: Write the given differential equation: dydx=2x \frac{dy}{dx} = 2x . Step 2: Integrate both sides with respect to x x : ∫dydx dx=∫2x dx. \int \frac{dy}{dx} \, dx = \int 2x \, dx. Step 3: Perform the integration: y=x2+C, y = x^2 + C, where C C is the constant of integration. Step 4: Apply the initial condition y(0)=3 y(0) = 3 : 3=02+C  ⟹  C=3. 3 = 0^2 + C \implies C = 3. Step 5: Substitute the value of C C back into the equation: y=x2+3. y = x^2 + 3. Step 6: Conclusion. The solution to the differential equation dydx=2x \frac{dy}{dx} = 2x with initial condition y(0)=3 y(0) = 3 is y=x2+3 y = x^2 + 3 . Solved on Solvelet with Basic AI Model
Some of the related questions asked by Sebastian Roberts on Solvelet
1. Use an appropriate solver to find the solution to the differential equation y′′−2y′+y=0 y'' - 2y' + y = 0 subject to initial conditions y(0)=1 y(0) = 1 and y′(0)=0 y'(0) = 0 2. Solve the partial differential equation ∂u∂t=k∂2u∂x2 \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} describing heat conduction in a rod of length L L with boundary conditions u(0,t)=u(L,t)=0 u(0, t) = u(L, t) = 0 .,
DefinitionDifferential equations are equations that express the relationship between the function in some variable and any of its derivatives. There are many methods and algorithms for solving the differential equations, such as separation of the variables and the integrating factor and a power series solution. There are also numerical methods for solving the differential equation, such as the Eulers method, Runge-Kutta method, and finite differences. These solvers are an essential tool in physics, engineering, and real-world modeling, as they allow us to predict the behavior of dynamical systems. For example, we can solve the first-order ordinary differential equation by integral both sides of it and then evaluate the constant of intergraion to get.
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