Example Solutions for

Step-by-Step Differential Equations

Separable Equations

See how to solve separable differential equations step by step:

solve y' = y^2 x
y'(x) = (x + 2) e^(-y(x)), y(0) = 0
sec(y(t)) y'(t) + sin(t - y(t)) = sin(t + y(t))
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First-Order Linear Equations

Solve first-order linear differential equations:

y'(t) - 2y(t) = 3 e^(2t)
x y'(x) - 4 y(x) = x^6 exp(x), y(1) = 0

See the steps for using Laplace transforms to solve an ordinary differential equation (ODE):

solve y'(t) - 3y(t) = delta(t - 2), where y(0) = 0
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First-Order Exact Equations

Solve exact differential equations step by step:

(3x + 2y)y' + 2x + 3y = 0 where y(0) = 2
solve t + arctan(y(t)) + (t + y(t))/(1 + y(t)^2) y'(t) = 0

Use an integrating factor to transform an equation into an exact equation:

2 t exp(2y)y' = 3 t^4 + exp(2y)
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Bernoulli Equations

Explore the steps to solve Bernoulli equations:

y'(x) - y = e^x y^2
x'(t) = x(t)(t x(t)^3 - 1)
solve y' + 5y = x y^4
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First-Order Substitutions

Solve a first-order homogeneous equation through a substitution:

solve x y' = y*(log(x) - log(y))

Learn the steps to make general substitutions:

solve 2 t^3 y'(t) = 1 + sqrt(1 + 4 t^2 y(t))
y'(x) = (1-x cos(y(x))) cot(y(x))
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Chini-Type Equations

Solve a Riccati equation step by step:

x^2 v'(x) + 2 x v(x) = x^4 v(x)^2 + 4
solve y' = y^2/x^2 - y/x + 1, y(1) = 0

Solve an Abel equation of the first kind with a constant invariant:

y'(x) = e^(2x) x y(x)^3 - y(x) - x e^(-x), y(0) = 0

Solve a Chini equation with a constant invariant:

2 x'(t) + t = 4sqrt(x(t))
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General First-Order Equations

See the steps for solving Clairaut's equation:

y(x) = x y'(x) + y'(x)^2

Solve d'Alembert's equation:

x(t) = t x'(t)^2 + x'(t)

See how first-order ODEs are solved:

solve y' = 2((y + 2)/(x + y - 1))^2, y(1) = 0
t y(t) (1 + t y(t)^2) y'(t) = 1
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Second-Order Constant-Coefficient Linear Equations

Solve a constant-coefficient linear homogeneous equation:

x''(t) = -k x(t)
solve y''(t) + 5y'(t) + 6y(t) = 0, y(0) = 1, y'(0) = 0

Explore how to solve inhomogeneous constant-coefficient linear equations:

solve y''+ y = sin(2x)
x'' - 2x' - 8x = 3e^(-2t), x(0) = 0, x'(0) = 1

See the steps for using Laplace transforms to solve an ODE:

y''(t) + 2 y'(t) + 2 y(t) = cos(t) delta(t - 3 pi), y(0) = 1, y'(0) = -1
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Reduction of Order

Reduce to a first-order equation:

t x''(t) - 2 x'(t) = 10 t^4
y''(x) + y'(x)^2 = 0

Derive the equation of a catenary curve step by step:

solve v''(x)^2 = (1+v'(x)^2), v(0) = 1, v'(0) = 0
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Euler–Cauchy Equations

Solve Euler–Cauchy equations:

solve x^2 y''(x) - x y'(x) + y(x) = 0
x^2 y'' - y = 0
2t^2*y'' + t*y' - 3*y = t, y(1) = 0, y'(1) = 1
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General Second-Order Equations

Investigate the steps for solving second-order ODEs:

t y''(t) - t y'(t) + y(t) = 2, y(0) = 2, y'(0) = -4
y'' - 2 cot(x) y' + (1+2cot(x)^2) y = 0
y''(x) + tan(x) y'(x) + sec(x)^2 y(x)==0
x^4*y*y'' + x^4*y'*y' + 3*x^3*y*y' = 1
solve y''(t) + sin(y(t)) = 0
x^2y'' + xy' + (x^2-1/4)y=0
solve (1-x) y'' + x y' - y = 0
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Higher-Order Equations

See the steps for solving higher-order differential equations:

solve y''''(x) + 16y(x) = 0
y''' - 2y'' + y' = 2 - 24e^t + 40e^(5t), y(0) = 1, y'(0) = 0, y''(0) = -1
y''' - y'' + y' - y = cosh(x)
y''''''(t) - 4y'''''(t) + 7y''''(t) - 4y'''(t) - 4y''(t) + 8y'(t) - 4y(t) = 0
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