Question

Introduction to differential equations

1. y' = x-1+xy-y

2. x^2 y' - yx^2 = y

Answer #1

Solve the following non-linear differential equations.
y'=xy''-x(y')^2

Solve the following differential equations through order
reduction.
(a) xy′y′′−3ln(x)((y′)2−1)=0.
(b) y′′−2ln(1−x)y′=x.

Homogeneous Differential Equations:
dy/dx = xy/x^(2) - y^(2)
dy/dx = x^2 + y^2 / 2xy

Given the second-order differential equation
y''(x) − xy'(x) + x^2 y(x) = 0
with initial conditions
y(0) = 0, y'(0) = 1.
(a) Write this equation as a system of 2 first order
differential equations.
(b) Approximate its solution by using the forward Euler
method.

Use the method for solving homogeneous equations to solve the
following differential equation.
(9x^2-y^2)dx+(xy-x^3y^-1)dy=0
solution is F(x,y)=C, Where C= ?

Solve the following differential equations with initial
conditions:
xy'-y=3xy1/2

solve differential equation ((x)2 - xy +(y)2)dx - xydy
= 0
solve differential equation (x^2-xy+y^2)dx - xydy =
0

Solve the following Bernoulli equations:
1) (2+x^2)y' +xy=x^3y^3
2) y′(x) + y/(x-2) = 5(x − 2)y^(1/2)

Differential Equations
Use principle of superposition to find a particular solution
x^2y''- xy+y =2x^3

differential equation one solution is given,
xy''-(2x+1)y'+(x+1)y=x^2; y_1=e^x

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