Linear Equations and Those Reducible to Them

Find the general integral of the equation $xy'+2y=3x$; find the particular integral that equals $y_{0}=0$ at $x_{0}=0$.

Find the general integral of the equation $xy'+3y=x^{2}$.

Find the general integral of the equation $y'+ay=e^{mx}$.

Find the general integral of the equation $(1+x^{2})y'-2xy=(1+x^{2})^{2}$.

Find the general integral of the equation $y'+2xy=2xe^{-x^{2}}$.

Find the general integral of the equation $y'\sin x-y=1-\cos x$.

Find the general integral of the equation $y'+y\cos x=\sin x\cos x$; find the particular integral that equals $y_{0}=1$ at $x_{0}=0$.

Find the general integral of the equation $(1-x^{2})y'+xy=1$; find the particular integral for which $y=1$ at $x=0$.

Solve the equation $(y^{2}-6x)y'+2y=0$, taking $y$ as the independent variable and $x$ as the function (the equation turns out to be linear in $x$).

Solve the equation $(x-2xy-y^{2})y'+y^{2}=0$, taking $y$ as the independent variable and $x$ as the function (the equation turns out to be linear in $x$).

Find the general integral of the equation $xy'+y=y^{2}\ln x$.

Find the general integral of the equation $y'+2xy=2x^{3}y^{3}$.

Find the general integral of the equation $3y^{2}y'-ay^{3}=x+1$.

Find the general integral of the equation $y^{n-1}(ay'+y)=x$.

Find the general integral of the equation $dx+(x+y^{2})\, dy=0$.

Find the general integral of the equation $(xy+x^{2}y^{3})y'=1$.

Solve the equation $(x^{2}+y^{2}+1)\, dy+xy\, dx=0$ (reduces to a linear equation by a simple change of variable).

Solve the equation $(x^{2}+y^{2}+2x-2y)\, dx+2(y-1)\, dy=0$ (reduces to a linear equation by a simple change of variable).

Solve the equation $y'-1=e^{x+2y}$ (reduces to a linear equation by a simple change of variable).

Solve the equation $xy'+1=e^{y}$ (reduces to a linear equation by a simple change of variable).

Solve the equation $y'\cos y+\sin y=x+1$ (reduces to a linear equation by a simple change of variable).

Solve the equation $y'+\sin y+x\cos y+x=0$ (reduces to a linear equation by a simple change of variable).

Solve the functional equation $\int_{0}^{1}\varphi (\alpha x)\, d\alpha =n\varphi (x)$.

Find $y$ from the equation $\int_{a}^{x}xy\, dx=x^{2}+y$.

Find $y$ from the equation $x\int_{0}^{x}y\, dx=(x+1)\int_{0}^{x}xy\, dx$.

Find $y$ from the equation $\int_{0}^{x}\sqrt{1+y'^{2}}\, dx=2\sqrt{x}+y$.

Solve the equation $2\int_{0}^{x}y\sqrt{1+y'^{2}}\, dx=2x+y^{2}$.

Prove that the differential equation

where $f_{1}(x,y)$, $f_{2}(x,y)$, $\varphi (x,y)$ are homogeneous functions, with $f_{1}$ and $f_{2}$ of the same degree, reduces to a Bernoulli equation after introducing the new variable $z=\dfrac {y}{x}$.

Solve the equation $(x+3y)\, dx+2y\, dy+a(x+y)(x\, dy-y\, dx)=0$.

Solve the equation $y^{2}(x+a)\, dx+x(x^{2}-ay)\, dy=0$.

Solve the equation $(x^{3}-xy^{2})\, dx+2x^{2}y\, dy-(x\, dy-y\, dx)=0$.

If $R$ is the resistance, $L$ is the self-induction coefficient, and $V$ is the voltage, then the current strength in the circuit $I$ satisfies the equation $L\dfrac {dI}{dt}+RI=V$. Find the current strength $t$ seconds after switching on, if $V$ is constant and at $t=0$ the current strength was $I_{0}$.

The same question as in the previous problem (find the current strength $t$ seconds after switching on), for $V=V_{0}\sin 2\pi nt$.

Find the curve for which the area of the trapezoid formed by the coordinate axes, the tangent, and the ordinate at the point of tangency is equal to $a^{2}$. Single out the curve passing through the point $(a,a)$.