Example 7. Integrate the Lagrange differential equation $y=\frac{3}{2} x y^{\prime}+e^{y^{\prime}}$.
Solution. Let's replace $y^{\prime}=t$. The equation takes the form $y=\frac{3}{2} x t+e^{t}$. This equality is differentiable with respect to $x$, then instead of $y^{\prime}$ we substitute $t$: $y^{\prime}=\frac{3}{2} t+\frac{3}{2} x t^{\prime}+e^{t} \cdot t^{\prime} ; t=\frac{3}{2} t+\frac{3}{2} x t^{\prime}+e^{t} \cdot t^{\prime} ;-\frac{t}{2}=\left(\frac{3}{2} x+e^{t} \cdot t^{\prime}\right)$. Since $t^{\prime}=\frac{d t}{d x}=\frac{1}{\frac{d x}{d t}}=\frac{1}{x^{\prime}}$, the last equation can be represented as a linear differential equation with respect to the unknown function $x: x^{\prime}=-3 \frac{x}{t}-2 \frac{e^{t}}{t}$. We solve this equation by the substitution $x=u \cdot v, x^{\prime}=u^{\prime} v+u v^{\prime}$. We get: $u^{\prime} v+u v^{\prime}=-\frac{3 u v}{t}-\frac{2 e^{t}}{t} \Leftrightarrow \Leftrightarrow$ $\Leftrightarrow\left\{\begin{array}{l}\frac{d v}{v}=\frac{-3 d t}{t}, \\ u^{\prime} \cdot v=-\frac{2 e^{t}}{t}\end{array} \Leftrightarrow\left\{\begin{array}{l}v=\frac{1}{t^{3}}, \\ u=\left(-2 t^{2}+4 t-4\right) e^{t}+\frac{C}{2} .\end{array}\right.\right.$

From this system, we obtain the general solution of the linear equation: $x=\frac{2}{t^{3}}\left(C-\left(t^{2}-2 t+2\right) e^{t}\right)$.

We add this equality to the expression for $y$ and obtain the general solution of the original Lagrange equation in parametric form.

Answer. $\left\{\begin{array}{l}x=\frac{2}{t^{3}}\left(C-\left(t^{2}-2 t+2\right) e^{t}\right), \\ y=\frac{3}{2} t x+e^{t} .\end{array}\right.$

Note. In the last equality, we can substitute the expression for $x$ from the first equality.

## Exercises

Integrate the equations not resolved with respect to the derivative.

1. $y^{\prime 2}-\frac{2 y+x^{2}-x}{x} \cdot y^{\prime}+\frac{(y-x)\left(y+x^{2}\right)}{x^{2}}=0$.
2. $y \cdot y^{\prime 2}+(x-y) \cdot y^{\prime}-x=0$. 3. $x^{2} \cdot y^{2}+3 x y \cdot y^{\prime}+2 y^{2}=0$.
3. $y^{\prime 3}+(x+2) e^{y}=0$. 5. $y^{\prime 3}-y \cdot y^{\prime 2}-x^{2} \cdot y^{\prime}+x^{2} y=0$.
4. $y^{\prime 2}-\left(\frac{1-x^{2}}{x y}+\frac{y+y^{3}}{x}\right) y^{\prime}+\frac{\left(1-x^{2}\right)\left(y+y^{3}\right)}{x^{2} y}=0$.
5. $y=y^{\prime 2}+x y^{\prime}-x$. 8. $x=y^{\prime 2}+\frac{y}{y^{\prime}}$. 9. $2 y^{\prime}+7 y^{\prime 2}=x$.
6. $y={y^{\prime}}^{2} \cdot e^{y^{\prime}}$. 11. $y=y^{\prime} \ln y^{\prime}$. 12. $x=\ln y^{\prime}+\sin y^{\prime}$.

Integrate the Lagrange and Clairaut equations.
13. $y=x y^{\prime 2}+y^{\prime 2}$. 14. $y=x\left(1+y^{\prime}\right)+y^{\prime 2}$. 15. $y=2 x y^{\prime}+y^{\prime 2}$.
16. $y=y y^{\prime 2}+2 x y^{\prime}$. 17. $y=x y^{\prime}-\frac{1}{y^{\prime 2}}$. 18. $y=x y^{\prime}+\frac{1}{y^{\prime}}$. 19. $x y^{\prime}+$ $+y^{\prime}=y .20 . x y^{\prime}+\sqrt{1-y^{\prime 2}}=y$.

## Answers

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