First order differential equations

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Problem: Find the general solution of the first order inhomogeneous differential equation: \[ y^{\prime} \cos x+y=1-\sin x \text {. } \]

8.1.1.1 First order differential equations

1.78 $

Problem: Find the general solution of the first order inhomogeneous differential equation: \[ y^{\prime}+\frac{x y}{4+x^{2}}=\frac{1}{\sqrt{4+x^{2}}} \text {. } \]

8.1.1.2 First order differential equations

1.78 $

Problem: Find a particular solution to the first order inhomogeneous differential equation: \[ y^{\prime}-\frac{y}{x+1}=e^{x} \cdot(x+1), y(0)=1 . \]

8.1.1.11 First order differential equations

1.27 $

Problem: Find the general integral of the differential equation. (Present the answer in the form of \( \varphi(x, y)=C) \). \[ (x+1) y^{\prime}+y(y+1)=0 . \]

8.1.1.20 First order differential equations

1.27 $

Problem: Find the general integral of the differential equation. \[ x y^{\prime}=\frac{y^{2}}{x}+y+x \text {. } \]

8.1.1.21 First order differential equations

1.27 $

Problem: Find the general solution of the linear inhomogeneous differential equation with constant coefficients. \[ x y^{\prime}=y-2 x^{2} \text {. } \]

8.1.1.22 First order differential equations

1.02 $

Problem: Find the general integral of the differential equation: \[ \left(2 x-1-\frac{y}{x^{2}}\right) d x-\left(2 y-\frac{1}{x}\right) d y=0 . \]

8.1.1.23 First order differential equations

1.78 $

Problem: Solve the Cauchy problem: \[ \left\{\begin{array}{l} y^{\prime}-\frac{y}{x}=x y^{2} \\ y(1)=\frac{2}{3} \end{array}\right. \]

8.1.1.24 First order differential equations

1.27 $

Problem: Find the particular solution of the equation with separable variables, satisfying the initial condition: \[ \left(1+y^{2}\right) d x-x y d y=0, \quad y_{0}=1 \text { when } x_{0}=2 . \]

8.1.1.27 First order differential equations

1.02 $

Problem: Find the general solution of the first order inhomogeneous differential equation: \[ x y^{\prime}+y=\ln x+1 \text {. } \]

8.1.1.28 First order differential equations

1.53 $

Problem: Find the general solution of the first order differential equation: \[ y^{\prime} \sin ^{2} 5 x-y^{2}+5=0 . \]

8.1.1.32 First order differential equations

0 $

Problem: Find the general solution of the differential equation: \[ \frac{d y}{\sin 7 x}+\cos ^{2} 5 y d x=0 \text {. } \]

8.1.1.35 First order differential equations

0 $

Problem: Find the general solution of the differential equation: \[ y^{\prime}=\frac{y}{x}+\cos ^{2} \frac{y}{x} \text {. } \]

8.1.1.36 First order differential equations

1.53 $

Problem: Find the particular solution of the differential equation: \[ y^{\prime} x \cos y+x \sin y=\sin y, y(1)=\frac{\pi}{2} . \]

8.1.1.38 First order differential equations

1.27 $

Problem: Find the particular solution of the differential equation: \[ \frac{y d y+z d z}{\sqrt{y^{2}+z^{2}}}+\frac{y d z-z d y}{y^{2}}=0, \quad z(1)=0 . \]

8.1.1.39 First order differential equations

2.55 $

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