Exact & Linear Differential Equations - Examples & Applications

Exact Differential Equations

Exact DE Problem (Basic)

Find the general solution for:
(2xy−3x2)dx+(x2−2y)dy=0(2xy - 3x^2)dx + (x^2 - 2y)dy = 0

Exact DE Solution

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Exact DE Problem (Intermediate)

Solve the following initial value problem:
(cos⁡xcos⁡y+2x)dx−(sin⁡xsin⁡y+2y)dy=0,y(0)=π(\cos x \cos y + 2x)dx - (\sin x \sin y + 2y)dy = 0, \quad y(0) = \pi

Exact IVP Solution

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Exact DE Problem (Advanced)

Find the general solution of:
(yx+6x)dx+(ln⁡x−2)dy=0,x>0\left(\frac{y}{x} + 6x\right)dx + (\ln x - 2)dy = 0, \quad x > 0

Advanced Exact DE Solution

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Non-Exact Equations and Integrating Factors

Integrating Factor Problem (Case 1: depends on x)

Solve the differential equation:
(3xy+y2)dx+(x2+xy)dy=0(3xy + y^2)dx + (x^2 + xy)dy = 0

Non-Exact DE Solution (x-dependent IF)

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Integrating Factor Problem (Case 2: depends on y)

Solve the differential equation:
2x dx+(x2+2y) dy=02x \, dx + (x^2 + 2y) \, dy = 0

Non-Exact DE Solution (y-dependent IF)

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Linear First-Order Equations

Linear DE Problem (Basic)

Solve the linear differential equation:
dydx+2y=e−x\frac{dy}{dx} + 2y = e^{-x}

Linear DE Solution

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Linear DE Problem (Intermediate)

Solve the differential equation:
xdydx−2y=x3cos⁡x,x>0x \frac{dy}{dx} - 2y = x^3 \cos x, \quad x > 0

Intermediate Linear DE Solution

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Linear DE Problem (Advanced)

Solve the initial value problem:
(x2+1)dydx+3xy=6x,y(0)=4(x^2 + 1)\frac{dy}{dx} + 3xy = 6x, \quad y(0) = 4

Advanced Linear DE Solution

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Bernoulli's Equation

Bernoulli Equation Problem (Basic)

Solve the Bernoulli equation:
dydx+1xy=y2\frac{dy}{dx} + \frac{1}{x}y = y^2

Bernoulli Equation Solution

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Bernoulli Equation Problem (Intermediate)

Solve the differential equation:
xdydx+y=x2y2x \frac{dy}{dx} + y = x^2 y^2

Intermediate Bernoulli Solution

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Bernoulli Equation Problem (Advanced)

Solve the differential equation:
dydx+y=xy3\frac{dy}{dx} + y = xy^3

Advanced Bernoulli Solution

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